It's been a few years since I heard this pronounced aloud, but my old undergrad prof's pronunciation of "3^^^3" was "3 hyper5 3". The "hyper5" part refers to the fact that three up-arrows is pentation. Similarly, "x^^y" is "x hyper4 y", because two up-arrows indicate tetration.

In general, add 2 to the number of up-arrows, and that's the hyper number you'd use.

(I should mention that I've never heard it used by anyone other than him, so it might have been just his way of saying it, as opposed to the way of saying it.)