  We construct non-power words which have small image in SL(2; 22n) for each n.
In particular, the corresponding word maps are non-surjective. We also use this
to construct word maps whose values are precisely the identity and a single
equivalence class of elements of order 17. In the second part we construct
words which have image consisting of the identity and a single equivalence
class of elements in Alt(n) for all n for any equivalence class with support
size at most 10.
