Iji and The Spirit Engine 2 are some of my favorites. TSE2 in particular is an amazing side scrolling RPG with a well developed story. They also both free, so just give them a google and play them now so you don't remain a schmuck for the rest of your life for missing out on them.
I love Pathologic and Void. They're indie games by a Russian studio called Ice-Pick Lodge and they're extremely atypical games. Their games are rather weird and not that well polished, but the story is just utterly amazing and completely different from most games. In The Void you play as a mute person trapped in this eponymous place. The Void is, like the name suggests, a place of utter darkness populated by interconnected "rooms" which sometimes appear as islands floating eerily or just nonsensical constructions. What's clear is that The Void is coming apart and that everyone in it is going to die. There are two parties - The Brothers and the Sisters. The first are visually monstrous entities that obey a strict code that dictates how Color should be used and how The Void should be taken care of. The Sisters are beautiful and soft-spoken, sometimes to the point of perhaps being outright manipulative - that is up to you to decide though. They seem to be unable to leave their chambers and require Color to fill their hearts - which opens other rooms and has several implications in the story line that would be way too spoilerish. Your survival in the Void depends on your usage of Color. Color is your life, your mana, you spend it and invest it in certain things so you can have more later. But the Brothers are opposed to all usage of Color for reasons beyond your comprehension in the beginning. The game never tells you what to do and generally possess great aesthetics. You don't have to fight the Brothers, or help the Sisters - there is no stated objective other than surviving, though there are several endings to the game. It's got a very heavy philosophical backdrop with the entire Void being a metaphor for life and death. If you bother to think about your predicament in the game you will soon start asking all sorts of questions about the nature of the Void and how the main character got there. I'd like to link the Game Over cutscene to showcase the game's atmosphere a bit: Youtube
As for Pathologic, I'm in love with this game. It's about a rather isolated town, presumably on the edges of the Russian steppe, where an epidemic of Sand Plague breaks out. The town itself is incredibly eerie and it become apparent that few things are normal here or ever where even before the Sand Plague. Strange constructions such as the Polyhedron loom over the town, time seems to flow in a rather strange way and one of the inhabitants of the town was supposedly clinically immortal. For that reason one of the playable character, The Bachelor, Daniel Dankovski, a student of medicine arrives in town to investigate this claim that could be a powerful argument for his hypothesis on the nature of life and death. This initial quest is entirely replaced by a rabid race against time to save the town from total destruction from the Sand Plague and internal strife among the most powerful families. The gameplay itself, is again, boring, you have to walk soooo much in this game, but the story is just amazing. Again, you are given very little direction as to what you have to do and the conditions for 'winning' or 'losing' are not clear - besides the fact that you have to keep several people known as 'adherents' alive. The game plays as a sort of mix between puzzle solving, dialogue tree investigating, first person shooting, stabbing and lots of trying to make sense of anything that happens in the game. The great part is that a lot of the story is implied by the aesthetics of the town and the overall atmosphere way before anything is spelt out, giving you a sense that you're fighting an adversary beyond your comprehension - perhaps death itself, as the town falls into chaos and misery.
Other indie games you might want to check out are Analogue: A Hate Story if you like a rather original and intriguing VN, Hotline Miami is loads of fun and has an excellent soundtrack, Bastion is an all around great game, highly recommended.