NOTE: While I'm no expert in the matter by any means, should you know something else or wish to add to what I wrote, feel free! 
RAM upgrades, that's a question that seems to pop up plenty of times in the community. In the making of this thread, I hope to have an authoritative response to some of the questions regarding RAM upgrades.
The first step would be to assess whether it's actually possible and these are things to take into account:
- The maximum RAM your CPU and chipset support.
Know your CPU, it can be seen at the System Info for example the one in the machine of the following picture is an i7 10510U:

Now you want to check its specifications, Intel lists the maximum supported RAM while for AMD you may need to search WikiChip for example. In this case, we see the following at its Ark page:

Great! It seems the maximum supported memory is 64 GB and supports dual channel (we'll get to that later on, but basically it can access 2 sticks of RAM at the same time). As for the chipset... that's more complicated, not many manufacturers specify the chipset they have used for each machine. There are some programs though, like AIDA64, which are able to show you the maximum amount of memory the chipset supports, and luckily, its trial version will display that information for free:

- Whether the current RAM is soldered onto the motherboard and no socket is provided.
If it is, there's no easy way to upgrade it and you might as well assume you can't for all intents and purposes. There are a couple of ways to know, when buying the PC if it states that the RAM is "on-board" it means it ships with it soldered, but it doesn't tell you if there are sockets you can use to expand on it or not. To know that you have to either open the machine and look for a RAM socket like the one in the picture below, or search online for the insides of your particular model. No socket = no upgrade.

There is no way to know it authoritatively through software. In that particular machine, there's no RAM soldered onto the motherboard and there is only 1 slot, yet programs report 2 slots are available. Because of that, the loss of 1 slot and the chipset limitation, the 64 GB of maximum memory the processor supported already went down to 32 GB for that particular system. It's a pity, but there's nothing we, as users, can do (maybe complain to the manufacturers :P).
Now that we know there's room for improvement (in the first screenshot the laptop showed it had 16 GB),
how do we know what to buy? Well, it depends on what you have already installed.
HWiNFO,
CPU-Z, or
AIDA64 among other programs can shed some light into the matter. With the info we got about the supported memory types from the specifications of the processor we can make informed decisions (right click, open in a new tab for a better view):


What that system has is DDR4-2666, also known as PC4-21300, so out of the 3 types the processor supported know we know we need DDR4-2666 (LPDDR3 fits in a different receptacle and LPDDR4 is soldered onto the board).
Could we get faster memory? Say... 3200 MHz? Sure, but it won't go faster than what the processor supports which is 2666 MHz. It is good because if you can find cheaper sticks with faster clock speeds they are suitable too, and that's because of those timing profiles you see highlighted in the bottom part.
The first D in DDR stands for double, the first line are therefore the timings for the memory running at 2666 MHz. But you can see below timings for 2400, 2133 and 1866 MHz, meaning that the RAM stick the machine has can run at any of those. And if there are more than one module they'd go at the highest common timing supported (assuming they're compatible with each other, more on this later).
So far we know we need DDR4-2666 and a stick of 32 GB as maximum, but let's assume the machine had 2 sockets instead of one; that is, the capability of having 2 RAM sticks.
Could we get any other stick of 16 GB, DDR4-2666, and pair it with the one we have? In short, no.
Of those timings, the first one is called the CAS latency (15 for 2666, the first number in the line). It has to match. In fact, pretty much any timing has to match for the memory to work in dual channel mode (read: faster). If they're different they won't do. Think of it as a rhythm: one person claps once every 2 seconds and another claps every 3, there's no way they
sound in tandem all of the time.
Even then, there can be sub-timing issues or motherboard incompatibilities preventing the system from working in dual channel mode or even booting when both sticks are present,
if at all possible it's recommended to get matched pairs (those sticks that are sold as packs); only then it's guaranteed they can work in X channel (X being dual, quad, etc.). They may still not work with your machine though, which is why it's good to buy from vendors that allow returns or exchanges.
Another good option would be to get the exact same stick of RAM you have installed, buying it second hand. Don't think they're used, when I got that machine I upgraded from the 8 GB it had to 16 GB and I sold the 8 GB part; you can be almost sure things will work. If you want to do that, look for the "Module Part Number" in HWiNFO, for the example we're following that would be "KHX2666C15S4/16G".
What about matching on-board RAM with a stick to work in dual channel mode? Well, same thing, characteristics have to match and some are even more picky on what they want as companion. You could try searching the net for your particular model, there may be someone who has already upgraded and has reported their experience

.
There's another program that can give you more information on the kind of RAM you have installed,
Thaiphoon Burner, and the free version is all you need (you won't be overriding any profiles anytime soon). I mention it because of 2 things, to show you there are even more things that can be the source of incompatibilities, but also some things that can be good as well. Take a look at this screenshot:

See that physical organization? Those 2 ranks cause problems sometimes, there are machines that only like single ranked RAM (uncommon I think?). However, did you notice there are more than one CAS latency supported at 2666 MHz? That's a plus! That means the memory could be happy with another stick that is DDR4 2666 MHz CL16 for example, or CL18; but yep, when it comes to real world matching, aside from the educated guesses we can do thanks to all this data, it boils down to trial and error

.
I hope it helps! If you have any question, don't hesitate to ask!
FAQ & Answers
Thanks! I was noting an increase (or maybe it was always so) of threads asking for information on RAM upgrades and I thought having a full guide on the matter would be useful to direct people to. There will always be people who post without searching the community first, but at least it is written.
Kind of like giving someone a fish vs. teaching them how to fish, we will still help them choose the appropriate one, but they'd have a single place to learn about the matter.
Edited the content by translating from Portuguese to English
Acer-Samuel
It is an excerpt from a conversation about increasing RAM on an AIO device, but it goes to show that even if AIDA64 or whatever program states that a certain chipset maximum RAM is X, that may just be true only if all of the sockets are present. In other words, not even knowing the chipset can you be sure of what the maximum RAM is for your system, it would depend on the design the manufacturer made around that chipset.
If I ever get my hands on a 32 GB DDR4-2666 MHz stick of RAM, I'll plug that into the machine used for the example of the article to see what happens. It may very well not work because of this reason and the maximum would be 16 GB.
Thanks to @billsey for his remark and for allowing it to be posted here as well.
Hi! Could you tell us the full laptop model and its main components? For example, if 16 GB were the maximum for the processor of that component it might have been only if it were divided in 2 slots, since you have just 1 that could be the issue. It is complicated to know sometimes without access to the designs of the manufacturer, and there's no way we get our hands in those, that's why it boils down to trial and error.
But we're lacking a lot of info to be able to assess the situation properly, I wouldn't be able to make an educated guess on why it happens (you're not alone though, a simple search in Google shows hundreds of cases like yours).
You're welcome! And nope, you can upgrade it with more and retain some of the benefits. If my memory doesn't fail I had a different but similar unit like the one I used to write the thread (an A515-54, without the G) that had 4 GB soldered and a stick of 8 GB installed. Its total RAM was 12 GB and part of it was operating in dual channel mode, at least that's what was reported by some info and benchmarking utilities (you'd see speeds greater than those you can get on single channel).
Not all of the memory would work at that speed of course, I never made proper tests under that configuration, but my guess is that once 8 GB of memory are being used, the remaining 4 GB would perform in single channel mode.
I noticed that some V7-482PG ship with a 4th gen i7-4500U, which has a maximum limit of 16 GB and if you have 4 GB on board that stick would already put you over the limit. And even if you didn't have any on-board memory, if the maximum were 16 GB as well, it may just be possible to get there by using 2 sticks, not 1 of 16 GB like billsey noted.
All CPUs have limits I'm afraid, whether it's Intel, AMD, Qualcomm, etc. just like 32-bit applications can't access more than 4 GB of memory the design of the processor and its memory controller limits the amount of RAM it can address. That's why the first part of the guide was go get acquainted with the hardware, know what kind of limits it has and then we'll see if it can be improved.
For example, I have an even older model that has a Core 2 Duo inside and has 2 slots for DDR2 memory if I'm not mistaken, currently populated with 2 GB each. It may seem like a little, just 4 GB of RAM in total, but that's the maximum that memory controller can accept. So that's it for that laptop.
With AMD is harder to know the limit because it's not disclosed in their specifications (that I could see) and one has to go to alternate sites looking for it, but there certainly is.
It's too bad you couldn't update with that stick though, would have been nice.
Tinkerer
Just for discussion, I have a older notebook which is using a i7-2630QM cpu. According to the intel fact sheet, it has 16gb limit and max memory channels of 2. However, this notebook is a 17" version that gives it 2-hdd plug-ins and 4-memory slots for expansion. I just placed a 8gb pc-1600 memory module into one slot while the rest of three are installed with 4gb pc-1333 module. It works just fine. Somehow the graphics handling is faster my desktop computer installed with AMD A10-5800K cpu. I know these are very old unit and I am not using it for business, just for fun.
For example, an even older Intel i7 967 Extreme was a 3 channel desktop chip, I think it was post Sandy Bridge when the successor to those 3-channel things established itself with quad channel designs, up until then even chips made for servers had 3 channel controllers as well. Quite different than those made for consumers though because they were able to address insane amounts of memory (think hundreds of GBs of RAM).
One of the first quad channel server chips from Intel must have been the E5-2637, a dual core chip with SMP (HyperThreading, so 4 logical cores) with a quad-channel memory controller. I know it's comparing apples to oranges looking at the specs of those 3 (the quad mobile chip you have, that desktop extreme with 3 channels and the server chip with 4 channels), but look at the difference in bandwidth:
i7 2630QM (Sandy Bridge):
i7 967 Extreme (Bloomfield, Nehalem microarch I think, precedes Sandy Bridge):
E5-2637 (Sandy Bridge EP):
I'm guessing those 51.2 GB/s of the server chip were achievable when using DDR3-1600, memory clocks of course influence the bandwidth, but still, look at what the difference dual vs tri vs quad channel makes.
That's why it's interesting to go for 2x 8 GB sticks vs. 1x 16 GB stick on most systems nowadays since dual-channel architectures seem to be the norm, there are videos on YouTube that demonstrate the difference like the one I'm going to post next, but the difference not only applies to games, it applies to everything!
For gaming and video decoding it is specially interesting if the desktop/laptop in question only has available the graphics card that is integrated with the processor (unlike in that old video), because the main system memory becomes its video memory too (although if I'm not mistaken some AMD APUs used to integrate some video memory in the die too... I can't recall exactly).
But I digress as usual, going back to that 17'' laptop you had, it has a quad mobile chip, capable of dual channel and 4 memory slots. If we name the memory slots 1, 2, 3 and 4 and one has only 2 sticks to populate the thing, they ought to go to slots 1 and 3 (or 2 and 4, I think that worked too), no dual channel otherwise. At least that's how they're typically wired anyway, I'm sure there'd be an odd duck around, but that's the rule of thumb.
The 1600 MHz memory is downclocking to 1333 MHz for 2 reasons, to play nice with the other guys and also because that's the max the memory controller in that chip is able to handle. I'm wondering if you were to "play" with how the chips are plugged in, i.e. in which slots, if you'd be able to see any difference in bandwidth. There's a program, MaxxMem2, that should be able to show some you some numbers on the bandwidth you're getting in different configurations.
Aside from the fact that the we're comparing a mobile part to a desktop one, because they have different TDP and power consumption limits (which is probably the one of the reasons Apple will try to use ARM chips in their Macs, because they can up the power limits), does any of them have a dedicated card or are they using the ones integrated in the processor? Because AMD has always had the upper hand in there, they just perform better. iGPUs from Intel leave much to be desired.
I'm curious as to what they (Intel) is doing with their Xe branded iGPUs, if they're going to perform like they're claiming they will (and later on their discrete GPUs based on that); but for now if one is interested in light gaming and buying a laptop without any dedicated graphics card, AMD it is.