Games that left you in awe at then end or were tense to play?

Skyrim
Mountain
Fable

Portal 1

Obligatory "I wish I could play Undertale for the first time" comment. And also Portal 2. And I would also say Bioshock Infinite, but I didn't even finish it once.

SWOTR
Mount & Blade series
Town Of Light
Godfather 1 &2
Crusader Kings 2
King Of Dragon Pass
Populous
AC Black Flag
....
and many more

System Shock 2 definitely left me wanting more and feeling great. I wish I could play through that again for the first time.

World of Warcraft. The first time I marched into the end of molten core and came face to face with Ragnaros with 39 others. That was an amazing moment.

Elite on the CBM64. The first time I managed to get docking right the game just opened up for me and became something of an addiction and the adverts with the thug being mostly harmless and the granny being dangerous, they made me strive to be so much more.

Bully: Scholarship Edition, Valiant Hearts: The Great War, Ceville, The Witcher 3, Vampire: The Masquerade - Bloodlines, most of the Daedalic Entertainment games, etc. :P

Borderlands 1&2,The Darkness, Dishonored, Mass Effect 2, Max Payne 2, DARK SOULS, Darksiders 2 are the games that come to my mind that i wish i could get a second first playthrough...

Fallout, Fallout 2, Arcanum, Gothic 1, Gothic 2, independence war 2, Civilization series

Bioshock Infinite, Life is Strange, Telltales The Walking Dead, Telltales The Wolf Among Us

Off the top of my head: To The Moon, Bioshock Infinite, Dreamweb, The Beginner's Guide, Syndicate, The Walking Dead, Braid, VVVVVV, Turrican 3, Might & Magic IV, Undertale.

Danganronpa. Would love erasing my memories to play again.

This war of Mine, Skyrim, Fallout NV & The Witcher 2

Deponia series (especially Goodbye and later Doomsday), Fallout 1+2, Antichamber, Warcraft III, Epic Battle Fantasy games, Quantum Conscience, Spore, Portal 2, Beyond Good & Evil, Fable: The Lost Chapters, and maybe Thomas Was Alone (but not that much), to name a few.

Well the biggest surprise recently was Tales from the Borderlands. Bought it for almost full price, but didn't regret anything. The first episode was amazing, and when the new episodes got released, they just made me sure that it is the best Telltale game so far. I'm a fan of the Borderlands universe anyways, and couldn't imagine how they will make TftB enjoyable, but damn they did everything right. It's just perfect. I wish for a Season 2 :/