Some things were not supposed to be Turing-complete. This is a collection of such accidents.
Stuff which is somehow limited (stack overflows, arbitrary configuration, etc) is still considered Turing complete, since all "physical" Turing machines are resource limited.
Although they were initially not supposed to, C++ templates are Turing-complete. For proof look at this paper (pdf).
The page fault handling in X86 can be used to implement a simple machine. Basically, a page fault pushes a word to the stack, which might underflow generating another trap. This mechanism provides a "subtract and branch if less than or equal to zero" instruction. Enough to implement a Turing machine. Demo video Talk video
Magic: The Gathering
Magic is a card game. Apparently, the rules are complex enough to reach Turing-completeness.
HTML5 + CSS3
Maybe this one was intentional, but the complexity of the computers people build in Minecraft is impressive. video example
SQL is usually not considered to be Turing-complete. However, with the features Common Table Expressions and Windowing, SQL is Turing Complete. The proof is in these slides. Also, look a Mandelbrot in SQL:2008. A broader exploration of various SQL-turing-machines was done by Fabien Coelho.
The C preprocessor is only Turing-complete, if executed in a loop, which feeds the output to the input ad infinitum. An example has won the IOCCC 2001 contest. Look into the herrmann1 entry. Nonetheless included in this list for coolness.
Apache Rewrite Rules
Apache comes with the mod_rewrite plugin to rewrite URLs. These rules ultimately are a Turing complete string rewrite system. Though the default configuration sets very low limits to recursion.
A Pokemon game, which is finished in 1minute 36seconds. 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.
Scala Type System
Apparently, you can implement the SKI calculus (which is Turing complete) in Scala types. However, the compilers stack overflows at some point.
Little Big Planet
Yet another game, where you can build a basic computer. For example, Conways Game of Life.
Server Side Includes
The venerable mail server is known for its arcane configuration. Turns out the configuration is also turing complete.