Most recently that would be The Last of Us, and before that Far Cry 3. Over the 20 years, that have been a few games that have had that impact on me. As a kid that would have been Castle of Illusion and Power Rangers The Movie.
Moving on to the PS1, N64, and Gameboy era, that would have been Final Fantasy VII, The Legend of Zelda: Ocarina of Time, Oddworld Abes Oddyssey, Oddworld Abes Exodus, Metal Gear Solid, and Pokemon Red, Blue, and Yellow. Tombi would have been another one I suppose. I only got Tombi last year, however I got a demo of Tombi years ago, and I must of played through the entire demo 4 or 5 times. I loved that demo and knew I had to get it.
During the PS2, GC, and Xbox era it would have been Shadow of the Colossus, Final Fantasy X, Halo 2, Metal Gear Solid 2, Metal Gear Solid 3, Metroid Fusion, The Legend of Zelda: Minish Cap, and Dynasty Warriors 3. I'm sure there were a few others, but I can't think if them off of the top of my head. Metal Gear Solid Twin Snakes was a big one as well. I played that game through every mode, got every dog tag, beat it 10 or more times. Its the one game I have never got bored of.
For generation 7 consoles, it would have been Final Fantasy 7 Crisis Core, The Last of Us, Far Cry 3, Journey, Alan Wake, The Walking Dead, The Elders Scrolls V: Skyrim, Metal Gear Solid IV: Guns of the Patriots, Portal 2, and Castlevania Order of Ecclesia. There's probably a few more that I'm not thinking of, but there you have it.
For generation 8 thus far, there's been nothing.