Can we enumerate provably non-terminating functions?
Here is a simple non-terminating function:

while (true):
    do nothing


This certainly does not terminate.

On the other hand, it is not possible to enumerate all non-terminating functions.  The halting problem is recursively enumerable but not decidable, from which it follows that the complement of the halting problem is not recursively enumerable (if it were, the halting problem would be decidable).  See https://en.wikipedia.org/wiki/Recursively_enumerable_language and https://en.wikipedia.org/wiki/RE_%28complexity%29.