Some people play normal video games blind, by getting some help from others in the gaming community:

http://kotaku.com/5484361/with-the-help-of-friends-blind-man-beats-legend-of-zelda

Text adventures are one possibility.

Also, there are a number of iOS and Android games that are accessible:

http://www.afb.org/messageboards.aspx?MBFolderID=7&MBTopicID=8076

Also, here's a game like Rock Band:

http://www.polygon.com/features/2013/8/6/4550490/blind-games-rock-vibe

still in prototyping, but check this out: http://bitoutsidethebox.com/shabda/

I just made a fun audio game app for android phones. Open field echo sounder.

https://play.google.com/store/apps/details?id=air.com.smolenski.openfieldechosounder&hl=en

My brother just recently downloaded this audio game for iPhone/iPad. Basically, it's a mix of old style radio dramas and choose your own adventure books. It's quite interactive and requires you to choose your next steps while progressing through the game. You are a secret agent in an interactive fictional world (I won't spoil it for you). It's completely audio oriented and have speech recognition when you have to choose your next step. I've heard some good reviews from blind and sight impaired people so you might like it as well. The name of the game is Codename Cygnus and you can download it from the App store. Also, you can check the intro audio on game site http://www.codenamecygnus.com/

It may be outdated but my dad is blind and an avid gamer. he plays mostly MUDS which are like mmo's that are purely text based. And with his screen reader he has been able to navigate them and have a blast for many years!

A whole system? I'm sorry, there isn't one. I can list about one or two dozen 'games' that work with text to speech functions, but it's a tiny market variable. I am working with a programmer to make a ios and android app game rpg for market right now, but it's looking like it's a month away as is.
I play skyrim, dragon age, stanley periable and others like that myself with my wife's help. It's 'possible' just not wide spread and also depends on the community support in-game.