The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times".

An example would be the existence of an unbounded class of inaccessible cardinals.

Now there are many examples of large cardinals $\kappa$ whose existence guarantees an unbounded class of inaccessibles below $\kappa$.

So, my first question:

Is there any large cardinal such that "$\kappa$ exists" $\implies$ "There exists an unbounded class of inaccessible cardinals in V" ?

An even stronger example of this would be something like:

"There exists a large cardinal of Type A" $\implies$ "There exists an unbounded class of cardinals of Type A".
Is there any large cardinal with this property?

I know that rank-into-rank cardinals satisfy an "upward reflection" property, but that only happens $\omega$-many times.

Would a Reinhardt cardinal (in ZF rather than ZFC say) reflect upwards unboundedly many times ?