First things first, this is my personal ranking of the portables that I've personally played or in some cases maybe just played the games. This means that I'll be skipping the Lynx due to lack of familiarity and possibly a few others. Sorry.
Next backwards compatibility (and emulators where applicable) do NOT count. No GB games bumping up the GBA system or NES emulator bumping up the PSP. This is to keep things somewhat even. Games that are downwards compatible (Pokemon Gold or Dragon Warrior 1-2 being played on GB) are fair game due to them being programmed to work that way. There's one edge case scenario where I *WILL* include it solely because I never had the system that game was meant for, but it won't boost the system enough to warrant worrying about it one way or another.
Obviously laptops, netbooks, phones, etc won't count or they'd top the list. Likewise the Virtual Boy does not count either since it's only technically portable but requires a table to be played. Switch, Turbo Express and Nomad (even if I never had one, I still had Genesis and PC Engine games) will count. All of this being said, portable game systems are my favorite way to play games. Due to having gone through tons of surgeries, doctor's appointments and now taking care of two parents in their mid-70s I have great need for something to do while sitting around waiting.
My criteria for how a system ranks includes library, battery life, how easily used the system is, portability (smaller is ALWAYS better), whether it uses a proprietary battery pack or if you can use AA/AAA batteries, cost, back lighting (sometimes), how easy it is to see the screen (without mods), among other things. In one or two cases a smaller library can make up for the size of the library by having a phenomenal selection.
Turbo Express (by necessity includes Turbo Graphix/PC Engine): Battery life, bad screen, fragile, requires four AA batteries to get the horrible battery life and heavy. Overall good library, but WAY too expensive.
Nomad (by necessity includes Genesis): Great library due to having the whole library of the Genesis to choose from, sadly battery life is abysmal. Due to battery life, cost, amount of batteries needed, and the fact that it shares the library with a home console I'm ranking this last.
Game Gear: Pretty good library, but horrendous battery life. Screen was fuzzy (though there is a replacement screen that fixes this issue), sound wasn't great and it's bulk was only made worse by requiring 6 AA batteries. Bonus points for the TV adapter, radio adapter and the adapter to play SMS games. Sadly bonus points don't matter.
Playstation Vita (includes the non portable PSTV): This would rank towards the top if Sony hadn't screwed the whole thing up. Battery life was subpar, support was pulled vastly prematurely due to the cell phone market kicking into high gear, the completely uncalled for touch pad on the back and several other issues marred this system horrifically. On the other hand, the amount of PC games that got portable official versions really brings up this system. What really kills this system? The fucking memory cards. This being the only version of Adventures of Mana with buttons would bring the system up several places if it weren't for the memory cards. It hurts the system that badly in my opinion. Also one of three systems on the list with Terraria, so that's fun. Has lots of emulator support (unofficially of course) and runs PSP games that were on PSN (or hacked into the system). Shame a lot of PSP games never made it to PSN. Also, having a battery pack is made worse by how difficult it is to replace since Sony got paranoid about the ease of making the Pandora Battery to hack the PSP (that's my guess anyway).
Game Boy Color: Yes. It's this high up the list. Mostly due to the lack of exclusive GBC games. There's some really good ones (Zelda Oracles of Season/Ages, Dragon Warrior 3, Pokemon Crystal and a few others), but the rest of the library is small and full of trash. Great battery life. OK screen (definitely a candidate for getting a replacement screen though). Compatible with original GB library. This system was a stop gap system between GB and GBA, and it shows.
Wonderswan (includes Color and Crystal): Ungodly amazing battery life. Decent screen. But the library is small and never got released outside of Japan so any RPGs are in Japanese only. On the other hand, I don't need to read Final Fantasy 1 to be able to beat it. Judgement Silversword is an amazing game that's sadly over $600 at this point. Buy it on Steam. One of the smaller portables on the list (bigger than the Game Boy Micro but smaller than the Game Boy Pocket) and it only requires one AA battery. The versions of FF1, FF2, and FF4 were later ported to GBA.
Neo Geo Pocket (includes Neo Geo Pocket Color due to only having played one NGP game and it wasn't on an NGP): Great system with amazing battery life. Screen is too dark. Links up to Dreamcast for benefits to both systems. Some of the best portable fighting games of all time, several of which compete with arcade games for how fun they are. Sadly the library is tiny due to SNK pulling out of the market due to Nintendo blowing them out of the water. The best directional pad (for fighting games) of all the systems on this list. Without exception. Would have ranked below WSC if it hadn't been for SVC: MotM.
Playstation Portable (includes PSP Go): Amazing system. Until you start picking the ports out of the line up, then it's just a great system. Battery life is about where I start to question whether it's adequate or not. UMD was a horrific idea that thankfully got lost relevance once games started showing up on PSN too. PSP Go was a decent system if you had enough games on PSN to make it worth it, despite the significantly worse battery life. If you ran games on memory cards the battery life was pretty good. Running them off UMD meant loading times on top of seeking data from a battery based system. This was a horrible idea. Having a battery pack is a downside, especially since it's not easy to replace due to how old it is. Amazing emulation system.
DS (includes DS Lite, DSi, DSi XL): The original DS is a horrible design. I will die on that hill. The DS Lite is an improvement in damn near every way. DSi was Nintendo's first foray into portable systems having region locks, but thankfully there were no major DSi exclusive games (I don't recall if Pokemon Black/White 1-2 had issues with the DSi and region locking or not) so this isn't an issue. Comes into play next gen though. DS Lite has great battery life and far superior screens to the original DS. Sadly this comes at the expense of having a battery pack which is no longer made, unless you want to risk it with a cheap Chinese knock off. DS and DS Lite can play GBA games, with GBA games sticking out of the bottom of the DS Lite but not the DS (the one major flaw in the DS Lite). Has some emulators. Closing the lid on any of these systems puts the game into suspend. The native library is decent with a ton of shovelware. The hinge of the DS Lite is notoriously fragile as well.
3DS (includes 3DS XL, 2DS, New 3DS, New 3DS XL, New 2DS XL): What the hell was Nintendo thinking with the name New 3DS? That's what every one was asking, and probably still is. Good system, with a better library overall than the DS due to having a lot of indie titles. Battery life is just above the Vita sadly, even with the New 3DS XL. If the 3D function didn't have the dial to turn it off I would have ranked this below the DS. Far fewer games require touch screen or other gimmicks thankfully saving this system from all the shoehorned in features (like Castlevania Dawn of Sorrow). Another battery pack. Virtual Console could have been a bonus if it counted and if Nintendo hadn't treated it like an after thought. Great for emulation when hacked.
Game Boy (includes Game Boy Pocket/Game Boy Light): Horrible screen. Legendary library. These systems have among the best battery life ever. Original Game Boy was portable, yet not. It was better than the Lynx, Game Gear or Turbo Express in size, battery life, etc. The amount of amazing RPGs, action games, fighting games, etc on these systems is difficult to truly get across. The Game Boy Pocket sadly has pretty bad battery life, but still better than a lot of the systems on this list. Definitely needs a screen mod though. Trying to play this system at night is pointless unless you buy a light for it. Yes, there's a definite personal bias towards this system. I will not apologize for this. Has some of my favorite games of all time.
Game Boy Advance (includes Game Boy Advance SP/Game Boy Micro/Game Boy Macro (aka hacked up DS Lite)): I don't know why they released the GBA with such a horrible screen originally, but the SP and Micro fixed this. Great library, just simply stunning. Battery life will last a long, long time as well. Despite the quite short battery life, I daily carry my GBM with Everdrive. The only other system I'll daily carry is my New 3DS because of the emulators. GBASP 001 was better than the original system, but was completely blown out of the water by the GBASP 101. First systems with a clamshell (outside of the Game and Watch systems) and definitely more portable than a good portion of this list. Sadly the SP and GBM have battery packs, but the GBA has a great battery life on two AA batteries. The back light mod for the GBA helps immensely. Also of interest is that there's a way to mod a DS Lite into a GBA Macro if the top screen goes to hell technically adding it to the GBA line up, but that's unofficial and requires a bit of work. Has a few emulators with the best one being for NES.
Now you might look through this list and realize there's one system missing that I mentioned early on. It's not being saved for last. I just don't know where I want to rank it yet. The Switch has good battery life, the library is great, but the controls feel awkward, and it's entirely too large for me to be willing to carry it. Definitely above the Vita. Definitely a system I'm looking forward to hacking for emulation, because god knows Nintendo sure as hell doesn't want to give you the ability to buy Virtual Console games on it.
Favorite games per system (chronological order) (no ports or remakes):
Game Boy: Final Fantasy Adventure, Batman, Final Fantasy Legend 3, Super Mario Land 3: Wario Land, Wario Blast, Zelda Link's Awakening, Dragon Warrior Monster 2, King of Fighters 96 (including because the GB version is it's own beast), Revelations (my first SMT game), Star Ocean Blue Sphere, Castlevania 2: Belmont's Revenge, Kirby's Pinball Land, Metroid 2, Operation C, Gargoyle's Quest
Game Gear: Insufficient experience to pick one. I only recall playing Sonic and one or two other games I don't recall off hand, so Sonic wins by default... yay?
Turbo Express: Galaga 90, Dragon's Curse, lots of shmups
Game Boy Color: Zelda Oracle of Seasons/Ages, Magi Nation
Wonderswan: Judgement Silversword
Neo Geo Pocket: SNK vs Capcom Match of the Millennium, Gal Fighters
Game Boy Advance: Shining Soul 2, Pokemon Mystery Dungeon Red, Advance Wars, Fire Emblem, Castlevania Circle of the Moon, Castlevania Aria of Sorrow, Mega Man Zero series, Mega Man Battle Network series, and way too many others to mention
DS: Pokemon gen 4, Kirby's Canvas Curse, Castlevania Dawn of Sorrow, Castlevania Portrait of Ruin, Contra 4, Dragon Quest 9, Meteos, Space Invaders Extreme (also the PSP version), Pokemon Mystery Dungeon games
PSP: Phantasy Star Portable, Dissidia 012
3DS: Smash Bros, Hyrule Warriors, Mario Golf, Super Mario 3D Land, New Super Mario Bros 2
Vita: Adventures of Mana
Switch: Mostly indie titles at this point.
This is probably bordering on too long. Oh well...