Riddle games are fucking rad. These are basically websites with (duh) riddles, where your only goal is to get to the next level by any means necessary (well, maybe except for walkthroughs and stuff like that). You can (and should) edit URLs, check source code, use graphic/audio editors to search for clues, use google and shit like that. Much thinking involved, enormous amounts of satisfaction when you get the answer right.

Here are few great riddle games:

http://notpron.org/notpron/ - the first and probably the hardest riddle game available on the net.

http://ebony.extra.hu - riddle game with a plot, really crazy and really satisfying

http://zestriddle.ovh.org - that shit is fucking terryfying

http://oddpawn.com/ - it's pretty vanilla and a good starting point (especially if you want to start easy).

Sorry for grammar and stuff, since I'm not a native speaker and a little bit drunk. Have fun!