Express a number with a combination of arithmetic operations and a single number
If you have any number at all, say $N$, then you can write it as $\frac 8 8$ followed by $N-1$ occurrences of $+\frac 8 8$.

The difficulty of the problem lies in finding short sequences of 8s that will work. The best known way is to enumerate all the possibilities. 

For instance, with six 8s, write them out in a row and then insert operators between them and after them, to generate all valid reverse-Polish notation expressions which lead to a single result. 

As well as the usual arithmetic operators, include a “concatenate” one which, immediately following two 8s, produces 88. 

Enumerating all conceivable RPN strings is not hard: there can only ever be five dyadic operators. Filtering them to select just the valid ones is not hard either: probably the best way is to include that process in the process of evaluating each one.