What is the TAK function for?
You might use it to evaluate the performance of programming languages:

http://en.wikipedia.org/wiki/Tak_%28function%29

since recursive methods are pretty useful for benchmarking. In particular, deep recursion tests the speed with which a language can make method calls. This is important because modern applications have a tendency to spend much of their time calling various API functions.

You will also need it to solve ballot problems like this one:

Suppose A and B are candidates for office and there are 2n voters, n
voting for A and n for B. In how many ways can the ballots be counted
so that B is never ahead of A?


http://mathworld.wolfram.com/BallotProblem.html

or just to generate Takeuchi Numbers: 

0, 1, 4, 14, 53, 223, 1034, 5221, 28437, ...