In the article Voevodsky’s Univalence Axiom in Homotopy Type Theory, an example is given of how types are not like sets: the existence of a nontrivial (nonzero) type $X$ such that $X\rightarrow X\cong X + 1$. However, I am unable to find this counterexample in the references in the article. Does anyone know this example? Is it a special case of some more general universal property, like maybe a type $X$ with $Y\rightarrow X \cong Y +1$ for all $Y$?
I'm not ready to venture into homotopy type theory yet, but there is an example in topos theory, which might be transportable into the context you want. In the object-classifier topos, the generic object $U$ satisfies $U^U\cong U+1$. This is Example 1 in an old paper of mine, "Functions on universal algebras" [J. Pure Appl. Alg. 42 (1986) 25-28].