For me, the limit would be a reasonable amount of information for them to do the job entrusted to them, given the risk of whatever activity was being regulated. For instance, a commercial airline pilot must prove his proficiency and health as well as provide name and address. A .5 to 55 lb. drone pilot must provide name and address to get a number. He must then apply that number to his drone. I'm happy with that.
[FWIW, imagine how AMA fixed wing aircraft owners feel! For over 50 years they've had radio controlled models able to wreak all sorts of havoc. To the best of my knowledge, none have. There have been very rare accidents but overall the safety rating for their flights must be up in the 99.999% range. UAS pilots are not being singled out, these guys have to now register as well. If anyone has a right to cry foul, it's AMA members.]