Pokemon Yellow: The interesting point about this speedrun is the bug it exploits. Turns out the game logic itself is Turing-complete in the sense that you can rewrite the assembly itself with game actions. For example, someone turned the game into a MIDI player.
MediaWiki Templates: In MediaWiki you can define templates. Since they provide recursion, you can apparently implement lambda calculus.
Some things were not supposed to be Turing-complete. This is a collection of such accidents.
Current Music: Desire -- Under Your Spell ♬