Let $$ F \colon \mathrm{Ord} \longrightarrow \mathrm{Ord} $$ be a $\le$-increasing class function such that every uncountable cardinal lies in the image of $F$.
Q. Does this property have a commonly used name?
To motivate this a little, let's say for now that $F$ is a flat function iff it has the property above.
Lemma. Let $$ F \colon \mathrm{Ord} \longrightarrow \mathrm{Ord} $$ be a normal (i.e. strictly increasing and continuous) function such that $F(0) \le \aleph_1$ and such that for all $i \in \mathrm{Ord}$ $$ |F(i+1)| \le |F(i)| + \aleph_1. $$ Then $F$ is flat.
The proof is an easy exercise.
Example. Let $(\kappa_i \mid i \in \mathrm{Ord})$ be the increasing sequence of Silver-indiscernables for $(L;\in)$ and let $$ F \colon \mathrm{Ord} \longrightarrow \mathrm{Ord}, i \mapsto \kappa_i. $$ Then $F$ is a normal, flat function.