Just another random mathematical speculation I came up with. It's probably not all that strong in terms of growth rate, but I thought it was interesting.
Define a 'hyperprime' number as a prime number whose digits, when added together, sum to a prime number, and the digits of that number added together also sum to a prime number, etc. until it reaches a 1 - digit prime.
The basic Hyperprime counting function HP(x) = the xth hyperprime number after 1.
The recursive Hyperprime counting functon RHP(x) = HP(HP(HP(HP...(x)))...) iterated a number of times equal to the value of x. I could probably come up with more extentions but I'm tired.