Enumerator for the language w#w^R?
Here is a simple Turing machine segment that reads a binary word $w$ starting at a position immediately to the right of the delimeter symbol # and replaces it with $w$'s successor in the lexicographic ordering. Here, L,R,S indicate moving the read/write head Left, Right, or Stay. 

This simple and powerful (standard) trick avoids a brute-force enumeration by using the results of previous computations to produce future ones. If you're allowed multiple tapes, it's now quite easy to accomplish the task you defined. Otherwise, there are standard $k$-tape to $1$-tape conversions.