Are there any algorithms that run in (2 ↑ ↑ n)? I’m wondering whether there are any algorithms that use so much time that they must be represented using Knuth up-arrow notation.

Required: Use more than one up-arrow for time complexity.

Bonus points: 


Have the algorithm be useful.
Have the algorithm be useful and optimized
Yes! A simple automaton-based algorithm for answering first-order queries in Presburger arithmetic (and some extensions) has worst-case running time about $2 \uparrow\uparrow n$, where $n$ is the number of quantifier alternations in the query.   Despite this formidable worst-case running time the algorithm has been implemented and gives useful results (see Hamoon Mousavi's implementation here).