How many times can a p2p permit be extended for 90 days?

I believe only once, since there is only one signature/date field for the CLEO to sign and date for any extensions. 

Once, at the issuing authority's discretion.

THIS^^^^^ and keep in mind it expires 180 days after initial issue not 90 days after it's been renewed.

Once. I just did mine last week from State Police. Got it back in less than a week!! I was surprised. Back in the day (dating myself) they just typed over the old date on my permits. On Monday I got a whole new permit with new date.

its sad to think that in almost every other state in the union all you need to purchase a handgun is money and ID, nothing from the LEO's