MATL, 15 12 bytes
0`ZetG>~}x@q
Try it online! Or verify all test cases (slightly modified version to handle several inputs).
How it works
Starting with 0, apply iterated exponentiation until exceeding the input. The output is the number of iterations minus 1.
0 % Push 0
` % Do.....
On a different subject: that J language looks cool. It has verbs, adverbs, conjunctions and the like :-) I might take a look at it when I have some time