Most Influential RPGs

My Personal Journey of Discovery
The first role-playing game I encountered was the original three Little Brown Books in a White Box published by TSR in 1974. It puzzled me and excited me. As I unpacked it all and eventually discovered how I could play it, I absolutely fell in love with White Box D&D. It's combination of brevity and customization potential compares well to all later games and it remains my favorite RPG to this day.
White Box introduced adventure gaming and with it several new concepts in gaming. The player's character as a single alter ego representation of your participation in the game setting is as far as I know original to the adventure game later referred to as role-playing. Leveraging the fantastic fiction that I greatly enjoy, the adventure game draws on genre archetypes for character inspiration and allows players to game the stories that until RPGs one could only passively read about. The RPG puts you into the fictional setting and allows you to make decisions and imagine yourself as that hero/heroine who accomplishes great deeds. White Box establishes the underworld, or dungeon as a place of adventure and while not directly described in previous literature the idea draws inspiration from many scenes borrowed from several sources, perhaps mythology being foremost among them. It is after all, the hero's journey that is being played out at the RPG table.
Ken St. Andre says he wrote Tunnels & Trolls in reaction to having read the White Box rules and found them wanting in areas. Tunnels & Trolls is perhaps the second RPG I played and from it I first discovered that the fun I was having with a character adventuring in a fantastic setting of make believe could be had solo. I soon discovered that I didn't need a "solo adventure" specifically aimed at solo play to enjoy the hobby without a group. There are ways to play any system and any published adventure as a solo game. While group play remains my preferred way to enjoy the RPG hobby, I have spent many hours alone creating my own fun with RPGs.
 The Fantasy Trip written by Steve Jackson and first published by Microgaming combines the role-playing experience with tactical hex-based combat strategy. Early D&D is a "theater of the mind" type of game where the action is imagined with few if any physical representations used. Subsequent editions including 3e, 4e and Pathfinder have utilized miniature figures and detailed combat grid maps to assist players in pursuing a more tactical combat mechanic.
Call of Cthulhu was my introduction to investigative gaming and real role playing of a character. Until Sandy Petersen and Chaosium released Call of Cthulhu, I approached adventure gaming from mostly a wargame perspective. My games were about combat with some exploration and puzzle solving, but very little character development and dialogue. With CoC I started to think of the character as something more than my "playing piece" - something rather close to a character in a novel who has personality, beliefs, goals, quirks and aspirations distinct from my own. I also came to appreciate the appeal of horror in the tabletop RPG through playing CoC, which incidentally got me reading the works of H.P. Lovecraft. One of the positive side effects of the hobby is that it helps introduce many of us to literature that we might otherwise not come into contact with.
RuneQuest and its excellent supplement on religion Cults of Prax both also published by Chaosium broadened my concept of what is possible in the world of the RPG. Up until Cults of Prax while playing White Box and the Advanced Game I had "hand waved" religion in my games. Having little interest in TSR's treatment of the subject and finding religion mostly portrayed as practiced by evil cultists in the fantasy stories I read, I generally ignored the portrayal of deities and worship. RuneQuest is not your basic dungeon crawler game, however and encourages characters who interact within the fictional society of the setting - RQ's default setting is Glorantha a mythical place where the deities are active and cults membership is the primary source of magic and skill progression. RuneQuest took my imagination to new places outside the standard medieval European flavor of prior RPGs and taught me the potential of world building - namely that setting can define the game.
My brief story now jumps ahead several decades as I come to Basic Fantasy and the old school renaissance (OSR). Basic Fantasy by Chris Gonnerman has been my introduction to the OSR community and helped me discover that there are a whole lot of gamers who continue to play and design for the older style games that I played in the 1970s and '80s and which I still love. Basic Fantasy is written using the Open Game License (WotC), but adapts that d20 system to give us a game that plays much like the older B/X game, albeit with some modern updated mechanics. The OSR has been a source of great enjoyment for me these past few years.
 The OSR is teaching me many new things about a hobby I have participated in for decades. Lamentations of the Flame Princess is an OSR RPG that has had a profound effect on how I view and play all systems. LotFP treats each of its four classes as being specialized in one thing and focuses progression on improving that one best thing. In other words, Fighters fight and are the only class which gets better at fighting as they level up. Clerics and Magic Users acquire spells and Specialists improve their specialized skills such as thieving or forestry. None of the races has infravision or can see in the dark. The dark is scary and to be without a light source greatly hampers the adventurer. Alignment makes sense in LotFP and is useful. I like alignment, not as a personality straight-jacket, but as an eternal struggle between order and disorder. Clerics in LotFP are agents of Law and Magic Users are aligned with Chaos - if you think about it, that makes sense. Other character classes are neutral. LotFP does not include a list of monsters or magic items as the author, James Edward Raggi IV believes both should be unique to each setting and should have history and lore attached to them. The mantra that "less is more" is one of my take-a-ways from LotFP.
As a result of broadening my hobby exposure I am also exploring the smaller press independent games and have found many good ideas among those products. From Luke Crane's The Burning Wheel I am learning the advantage of scaling mechanics. There are at least three ways to adjudicate a fight in Burning Wheel depending on how how much time and effort you would like to devote to that encounter. One is less detailed and dramatic and a good way to decide the outcome of an unimportant, but violent encounter. A second involves more detail and skill use, but is less involved than the third which is aimed at producing a tactical exchange that heightens drama and tension and is more likely to produce that memorable heroic event that can help define a campaign. Burning Wheel also introduced me to the concept of "let it ride", in other words economize on the dice rolling. A die roll results in Burning Wheel stands until there is a significant change in circumstances. In other words if the character succeeds at a sneak roll, there is no need to call for another until something changes. Social conflicts in Burning Wheel are given similar weight to combat encounters in terms of having detailed mechanics for the use of argument strategies and changing an opponent's mind or winning a debate.
There are many games I enjoy which are not on this list and finding the time to play them all is much harder than finding one that I like and look forward to playing.