A back and forth prime type of problem

To illustrate what I have in mind, let's start with 137.

137 is a prime number. Take off the 1 leaves you with 37 which is another prime number and taking off the seven leaves you with 3 which is yet another prime.

By reversing what I did, you start with 3, then add in the 7 digit on the right-hand side then add in the one on the left-hand side, all prime numbers. Now maybe there's a digit you can add on the right-hand side of 137 say 137a, yielding another prime number. And then another digit you can add to the left-hand side yielding b137a which (hopefully) is yet another prime number.

The puzzle is to find the string of prime numbers, growing in the manner just illustrated, growing to the largest prime number that can be determined by the pattern I just illustrated (you don't have to start with 137, you have the choice of digits 1-9 on the left-hand side to grow with and the digits 1,3,5,7 and 9 to grow with on the right-hand side).

How far can you get?