Let $C_b^m(U, V)$ denote the space of not necessarily bounded mappings $g\colon U \to V$ that have $m\ge 1$ continuous and bounded Fréchet derivatives of order $1,\dots,m$. I see $C_b^m(U, V)$ reasonably often, which is endowed with the seminorm $|\cdot|_{C_b^m}$ defined as the smallest constant $C\ge0$ such that $$ \sup_{u\in U}\|D^m g(u)(x_1,\dots,x_m)\|_{V}\le C \|x_1\|_U\dotsb\|x_m\|_U\text{ for }x_1,\dots,x_m\in U. $$
Naive examples would be Nemytskii operators. I see if $U$ is a Banach space like $C_b(\overline{\Omega})$, then Nemytskii operator would do. But if $U$ is a Hilbert space, e.g., $L^2$ then differentiability implies that the Nemytskii operator must be generated by affine functions (e.g., Appell--Zabreiko's book, nonlinear superposition operators), which are not interesting. For smooth Sobolev spaces that are Banach algebras, the differentiability is fine but they won't be bounded?
I wish to know what are examples of non-trivial (or non-affine) elements in $C_b^m(H,H)$ with $H$ being an (infinite dimensional) Hilbert space.