I'm coming to this one a bit late after having the flu for a few days.
Reading through a lot of people's responses, there are the same regular names that keep popping up. A lot of those names have influenced me too, probably mostly because they won the fame lottery and happened to be in the right place at the right time. Whether they're actually better designers than others who are struggling away at the hobby, that's a matter of much debate. Whether they deserve to be as renowned as they are... well seriously, do the Kardashian's deserve to be famous? What do they actually do for society?
I'm going to try and keep the cult of personality out of this response, instead focusing on games, mechanisms or movements in the hobby. Also, most of the influences I've seen have been positive things to aspire to... there are also negative influences which indicate things to be avoided.
I played a few games before this, but TMNT was the first game to show me how much fun an RPG could be. It was also eye opening to see a game which wasn't set in a pseudo-medieval fantasy world, so this really opened my eyes to the potential of gaming.
Positive: Darksun and Planescape
Just as TMNT had shown that not all gaming had to be fantasy oriented, Dark Sun and Planescape showed me that not all fantasy gaming had to look like traditional fantasy. That blew me away, and became something I've tried to emulate ever since.
Negative: The Satanic Panic
Watching friends have their RPGs burnt, and having my own RPGs destroyed and thrown out as rubbish was a big influence. I never saw gaming as evil or satanic, just as a way of opening up the imagination and mind to new ideas. That's where I realised that organised religion is just a cult, fearful of new ideas and highly reactionary when they show. Roleplaying was my escape, my sticking it to the man, my punk.
Positive: Mage the Ascension
A perfect storm of philosophy, rebellion, postmodernism, and zen. Mage was the game that pushed my experiences to heights I've tried to reach again, but it always felt hamstrung by the Storyteller system. It's the game I've always wanted to perfect, but never managed to achieve.
Positive: the Big Model
Like all sociological theory, the Big Model was filled with words that didn't quite mean what they appeared to, and we all took something slightly different away from it. But the work at The Forge set a seismic shift across gaming, leaving some amazing stuff in its wake.
Positive: The Stockade
All the big stuff in gaming happened overseas. In the UK, or North America, or quirky stuff from Europe. Any time something started in Australia, it lasted a few months then faded away. The Stockade was the first attempt I was aware of for multiple Aussie designers to work together, on their projects but feeding back to one another and sharing resources. Then it collapsed...
I always thought that writing and "publishing" a hack was sheer laziness. If you're gping to write something, make it truly visionary, make it your own. Don't just tweak someone else's work and claim credit for it. As a result, I've generally avoided the Hack scene.
Since I joined in the platform's beta phase, the communities on G+ have been an amazing source of inspiration and influence. I've met so many people here a d have seen so many great projects develop.
Negative: Powered by the Apocalypse
I appreciate it in some ways, I hate it in others. I suspect that if it weren't for the cult of personality centred around it's founder, the whole thing would gave imploded by now. I think too many people are claiming too much from the Apocalypse engine, and most of it is just hype... I'll just stand over here doing my own thing.
Negative: The OSR
Another one of those perfect storms but this one works in reverse. We didn't have our roleplaying games destroyed in the 80s just so wanna-bes could write hacks that ramped up the trappings of satanism, occult nonsense, and in-yer-face controversy, expletives or other rudeness on the cover. This is the kind of bullshit that gives the hobby a bad name. Get over it, grow up, don't take us down again.