My favorite Turing tarpit is FRACTRAN, devised by John Conway. It's flabbergasting that it is actually Turing complete, but it is.

A FRACTRAN program is a list of fractions, rational numbers. You input an integer and then you find the first fraction that gives you an integer when multiplied by your input. That process is repeated with the new, multiplied number until there is no fraction that produces an integer when multiplied.

Conway wrote a program that generates primes, there's others that mathematical sequences, but my favorite is just the addition program:

    3/2

Given an input 2^x * 3^y, it will stop at 3^(x+y), which adds x and y together.