Impulse response of system with $\max\left\{\lvert x[n] \rvert\right\}\ge\max\left\{\lvert y[n]\rvert \right\}$
Not sure what exactly the question is: find some LTI system that meets this constraint (which is trivial) or create a testable condition for any LTI system. (Which is not trivial).
For the latter: Discrete LTI systems are guaranteed to not increase the maximum amplitude if the absolute sum of the impulse response is smaller than $1$, i.e.
$$\sum |h(n)| \leq 1 $$
The proof is a bit tedious, so I'll skip it unless there is need for it.