TITLE: "Bootstrapping" an unbounded class of inaccessible cardinals
QUESTION [8 upvotes]: 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 ?

REPLY [4 votes]: The maximality principle implies a sweeping uniform version of the idea you mention. 

Hamkins, Joel David, A simple maximality principle., J. Symb. Log. 68, No. 2, 527-550 (2003). ZBL1056.03028.

Namely, theorem 14 of that paper shows that if MP holds, then there is a proper class of inaccessible cardinals, if any; and similarly for Mahlo cardinals and indeed any large cardinal notion that is downward absolute to small-forcing grounds. 
To explain, the maximality principle is the principle that any statement that is forceably necessary is already true. That is, if there is a forcing extension such that the statement is true in all further forcing extensions, then the statement is already true. This principle is expressible in modal terms by the scheme $\Diamond\Box\varphi\to\varphi$, and indeed, the modal logic of forcing was introduced in this paper specifically because of the connection with the maximality principle. 
The point now is that if MP holds and there is an inaccessible cardinal, then there cannot be only one or a bounded number of them, because then we could collapse them all and thereby make it forcing necessarily true that there would be none; so under MP there would have to have been none to begin with. So if there is one, there must be a proper class of them.