Here's a looooong list of games I suggest, so get ready; I'll try to group these in some method to my madness, but it'll still be crazy. All of these games are available as PDF files or via Print on Demand.
I'm not terribly helpful when it comes to selecting a single game, but here's some stuff I find interesting. If you're looking at it as a terms of various systems being your primary focus, I recommend Eclipse Phase/The Laundry, Shadowrun/Vampire: The Masquerade, D6/Septimus, Remnants, Dust Devils, and Wu Xing/Part-Time Gods/The New Epoch, and Spellbound Kingdoms as general exemplars.
Against the Dead: I actually recommend this game for a number of reasons, but the main reason is that it's a good example of a lighter, easier game (for people already familiar with 3.5 OGL) that also shows how a system can be altered to handle a different setting, though Spycraft does much the same thing. (Good but not a super duper must play)
Bill Coffin's Septimus: I liked this because it's the sort of cerebral and alien science-fiction that engages me, but it's also free and based on the excellent d6 system. I'd check it out if for no other reason than that it's free and it's there (at least if the site's not lying to me), but I maintain that nobody will be disappointed by it. (Free, and decent, definitely worth trying, good for its system)
Blood Dawn: This is actually a cheap 80's knockoff game, but it's incredibly good, if one looks deep enough. It's edgy, conspiracy theory-esque, and kinda crazy, but it's a good example of how a game can be high-quality and fun without being a super duper pain. (Interesting, not a must play, but I enjoy it a lot)
Chronicles of Arax: Another free game that's really decent, I like it because it provides a choose-your-own-adventure feel and a lot of enjoyment can be had from it. It's free, too. (Interesting for being solo and for its own sake)
Corporation: Rather expensive, but a good game. It's got a lot of stuff between its covers and fills a niche that is occupied by the likes of Deus Ex and Syndicate in video games. I'm probably too much of a fan of the genre to speak objectively, but it's good stuff nonetheless. (System's not super duper awesome, but setting and gameplay are)
Classic Traveller: This is a bundle of the stuff you'll need to play Traveller [sic] with a group of people. It's a classic and it's great for insights as well as playing of its own accord. It's also cheap, and so long as you remember that it comes in three separate .pdf files (which causes an inordinate amount of confusion for some people) you'll find lots of enjoyment and great material in here. (Interesting if somewhat obtuse system)
D6: Ok, the whole set of core D6 books free? I love this system with an incredible passion for its simplicity (so long as you stick away from the arcane [ha!] supernatural powers sections), I'd say that it's a great example of a simple but enjoyable game system. It's free, too, which is icing on the awesome cake. (Same as Septimus, these tend to cut more to the chase, a favorite of mine)
Die Type System: This is actually immensely better than it sounds. Admittedly, while it does play exceptionally well, it is very much a parody of traditional JRPG video games, so be warned that it shows. However, that makes it even more endearing for some audiences, and it makes it an even better example of setting and system meshing together. (Interesting as a case study, also high quality, personally I love it for nostalgia and JRPG love)
Do: Pilgrims of the Flying Temple: Don't let its "kiddiness" fool you, Do is a very great game, with a very solid mechanical system under its whimsical exterior. It's not for all groups, but did I mention that John Wick was a contributor? It's also dice-free, which makes it a very interesting game for someone with a D&D and Shadowrun background like myself. (System is extremely interesting, game itself is high quality)
Dust Devils: While I don't really know that much about poker which limits my real enjoyment of this, it's another example of a diceless game (it uses cards, in case the opener about poker was too subtle) that is really, really good. Plus, it's indie and it's not too expensive. (Not very proficient with it, looks interesting)
Eclipse Phase: One of my favorites, and possibly my favorite setting to ponder, Eclipse Phase combines hard transhuman science-fiction, some really cool setting stuff, conspiracy theories, political intrigue, and pretty much every last aspect of the modern world into a post-apocalyptic scenario (hey, at least we have space, right?) and basically makes it one of the most awesome tabletop games ever. Just the setting alone is worth the price, and the system is pretty solid (if a predictable percentile one). (Super high-quality game, system's not actually too thrilling, a favorite of mine)
Eclipse: The Codex Persona: Free to trial "shareware" licensed tabletop game that's actually really, really awesome. I recommend it because it's much like Against the Dead in turning d20 on its head and doing a lot of stuff with it, though it focuses more on changing the system than the setting. Give it a look, after all if you don't like it it doesn't force you to pay for it. (Very interesting because it takes a familiar system and messes with it a good deal)
Grey Ranks: Handling the Holocaust and coming of age in an incredibly serious way while encouraging engaging and dramatic storytelling, Grey Ranks wins awards for a reason. Go check it out, I shouldn't have to say more. (Super game, very cool, system is very interesting)
Heavy Gear: This is one of my favorite games because it involves mecha action. I'll admit that my tastes are not super-duper sophisticated, but Heavy Gear is seriously one of my favorite games, and I love the integration of tactical and roleplaying gameplay that the Second Edition rules offer. (System's good, setting is awesome)
King Arthur Pendragon: If you love Arthurian myth, I have to recommend this game. I have a print copy myself, and I can vouch for how incredibly awesome it is. It's maybe less freeform than some games, but it's really, really an incredible show of composition and design. (Good for the things it covers, maybe a bit too control-freakish)
Legend of the Five Rings: This is just an incredibly awesome game on every level, with a great show of force on setting, art, and system. It may be incredibly expensive, but it makes up for it by being one of what I would consider the best games of recent years. (Great setting, art, and system, I love this game)
Oathbound Seven: Based off of Pathfinder, Oathbound Seven isn't so much exceptional for its system (which is admittedly fantastic), but for its incredible setting which is packed full of content. If Die Type System is inspired by the system of classic JRPG's, this is inspired heavily by their epic scale. I'm always in awe when I take a look at this, and at $10 for just under 500 pages, you get a lot for a steal! (Setting's good, system's a old standby, I love it nonetheless)
Outbreak: Undead: Also really, really awesome, Outbreak: Undead attempts to be a realistic zombie simulation, with an almost obsessive level of detail. It's gritty and gory, and it's just a really good game. Throw in Free Content Fridays, which means that you get essentially an ever-expanding game, and its $20 price tag gets a little bit sweeter. While I don't buy its claims to super realism, it's fun and it makes a great zombie game. (Interesting for how it handles things, system's pretty predictable, still one of my all-time favorites)
Part-Time Gods: Interesting game with a focus on magical realism. It focuses a lot on social intrigue and psychological stuff, akin to Vampire: the Masquerade. (System's simple, the setting and how they interact are cool)
Remnants: Post-apocalyptic feudal mecha combat? Please. This may be because I'm a fan of this sort of genre, as people may have noticed, but Remnants is both high quality and well written. (System's great for its speed and such, setting's stellar, definitely one of my favorites)
Shadowrun: Ok, I'm a bit of an edition purist here, but not only was Shadowrun my first tabletop game but I feel that its system is one of the best ones out there. I admit that I've never played anything but the third edition, and I don't care for fourth edition terribly, but I absolutely love it. (My absolute favorite game ever both in setting and system)
Spellbound Kingdoms: I love this game's setting, system, and most importantly its combat system. That's not to say that it's lacking in other ways, but its combat system is remarkably cool and focuses on building towards ends, making hand-to-hand combat more intellectual than just attacking and occasionally choosing a maneuver. (System and setting are fun, one of my favorites)
Stars Without Number: OSR-inspired science fiction game; free with supplements that you can buy. Very simple but also very good. It does exactly what it sets out to and is remarkably well focused; there's a reason it gets great reviews. Check it out. (Not super duper, but interesting for comparison to modern d20)
Stellar Wind: Warning, contains real (if somewhat altered) rocket science. It's a good game even if it happens to be a little obtuse at times, it's definitely more for people who enjoy math or physics but it's still pretty decent. (Interesting if you love math stuff, not so much for less math minded people)
The Laundry: Based off of a series of books and short stories I've never actually read, The Laundry mixes low-glamor spy business with Lovecraftian elements to make an outstandingly funny, engaging, and deep setting. Uses the same system as Call of Cthulhu, and is basically Lovecraft in a more modern day setting. Unique for the fact that you can summon eldritch abominations via your smartphone. (Still pretty cool, system's BRP, so it's nothing exceptionally mathematically difficult)
The Mutant Epoch: A great post-apocalyptic setting similar to Fallout or the likes, The Mutant Epoch is both deep and well thought out, with a massive amount of online content. My only gripe with it is that it tells way more than I'd like to know about prostitution in the post-apocalyptic era, but other than that it's good. (Decent quality, lots of content, including some you probably don't want)
The New Epoch: A steampunk game that comes in a number of books, I recommend it or its exceptional amount of stuff and just being rather high quality in general. It is unfortunately split up into a number of volumes, which becomes rather unwieldy and pricy very quick. (Good, d20 based with way too many modifiers but not too horrible)
Triune: Mixes religious and science fiction tropes and conventions to form an incredibly something game. I'm not really sure what to say about this game, but I'd say that if you're looking for something out of the ordinary but not plain bad, Triune can fill that role. (System's decent, setting goes all over the place and is very interesting)
Vampire: The Masquerade: There's a reason that Vampire: The Masquerade has such a giant presence in the roleplaying game community, and it's not just because it's every depressed teenager's angst trip. Its system and setting are well done and deep, and the book is made with high quality standards. (Setting's dark, system's really good and similar to Shadowrun's, which I love)
Wu Xing: The Ninja Crusade: I recommend this partly due to its martial arts system, but it's a great example of a high-quality game. It's very genre-heavy, which makes it a great example of how setting and system can be integrated. (Similar to the New Epoch and Part-Time Gods)
Disclaimer: I review for DriveThruRPG, who I link to incredibly often here. I also am an amateur game developer, hence my ranting about setting and system and their integration.