Personally I've enjoyed some of the Japanese indie games (they call 'em doujin and often rip characters straight from commercial works but it's still the same thing, independent game development) even though I don't understand a word of their story. Bunny Must Die [http://www.platinedispositif.net/games/bmd.htm] (kinda like Metroid or Metroidvania with the now ubiquitous time control features), Hitogata Happa [http://www.platinedispositif.net/games/happa.htm] (bullet hell shmup with kamikaze mechanic), Scarlett Weather Rhapsody [http://www.tasofro.net/touhou105/index.html] (fighting game with bullets but not the WarTech kind), PatchCon! [http://www.tasofro.net/patchcon/index.html] ("tower defense" but with mobile units instead of towers) and such but these are only sold at retail, not online and getting them outside of Japan without warezing them is difficult (well, order them at the right store [http://www.paletweb.com/] but still much less convenient than a download) and takes a while so they're probably not suitable for your journey (plus they tend to be designed more conventionally rather than the experimental kind of indie game you seem to be looking for). Considering I listed two games based on Touhou here maybe pointing at the Touhou main series [http://www16.big.or.jp/~zun/html/game.html] also makes sense...
I'll go dig into my list of free games for easier access...
Cho Ren Sha 68k [http://www2.tky.3web.ne.jp/~yosshin/my_works/index.html] (shmup)
Kenta Cho's games [http://www.asahi-net.or.jp/~cs8k-cyu/index_e.html] (mostly shmups, inspiration for the commercial game Blast Works)
Frequon Invaders [http://home.comcast.net/~arch.robison/frequon.html] (not Japanese but a great game to confuse onlookers)
Oh and how could I forget the evergreen Crimsonland [http://www.crimsonland.com/]... EDIT: And Sigma Team [http://www.sigma-team.net/] (I think they're indie), the creators of games like Alien Shooter, Alien Shooter 2 and Zombie Shooter...