In MK (Kelley-Morse) class theory, if i add an axiom that any cardinal except $On$ has an inaccessible greater than it (ie. essentially a Tarski/Grothendiek universe axiom), does that compel me to admit the existence of any other large cardinals (eg. measurable cardinals)?
Broadly what are the known side-effects of an ever increasing sequence of inaccessibles in MK?
For example this answer Is the axiom of universes 'harmless'? states that universes are used to resolve Fermats Last Theorem (although they can be dispensed with).
This has no noteworthy side-effects that I can think of. In particular, it does not imply the existence of any stronger kinds of large cardinals.
There's a nice list of large cardinal axioms ordered by their (consistency) strength at Cantor's attic. As you can see, the Grothendieck universe axiom is near the very bottom, being the first axiom on the list that is stronger than just the existence of a single inaccessible. The list is formulated for ZFC rather than MK, but MK only very slightly increases the strength (for instance, MK with a proper class of inaccessibles is still weaker than ZFC with a $1$-inaccessible cardinal in consistency strength).
In concrete terms, pretty much all the stronger kinds of large cardinal are automatically at least $1$-inaccessible (that is, they are inaccessible and are a limit of inaccessible cardinals). If $\kappa$ is the least $1$-inaccessible cardinal, then $V_{\kappa+1}$ gives a model of MK in which there is a proper class of inaccessible cardinals but no $1$-inaccessible cardinals.