A few comments:

a) this problem was mentioned in 2004 edition of R.Guy's book "Unsolved Problems in Number Theory". The author's comment indicated that at the time the solution was not known.

b) In 1997 Jerry Myerson reformulated original question about infinite sequence of right-extendable (truncatable) primes in the form presented here. He also mentioned that it was easily solved in bases 2 through 6.

c) In 2007 the record-holding sequence was only 14 numbers long.

It certainly seems that this problem (about infinite sequence) is still unsolved. The fact that we can only add digits 3 and 9 is pretty obvious because digits 1 and 7 cannot be used more than twice (modulo 3 argument proves that) - therefore we can assume that the sequence begins after the last time those digits were added.

A few comments:

a) this problem was mentioned in 2004 edition of R.Guy's book "Unsolved Problems in Number Theory". The author's comment indicated that at the time the solution was not known.

b) In 1997 Gerry Myerson reformulated original question about infinite sequence of right-extendable (truncatable) primes in the form presented here. He also mentioned that it was easily solved in bases 2 through 6.

c) In 2007 the record-holding sequence was only 14 numbers long.

It certainly seems that this problem (about infinite sequence) is still unsolved. The fact that we can only add digits 3 and 9 is pretty obvious because digits 1 and 7 cannot be used more than twice (modulo 3 argument proves that) - therefore we can assume that the sequence begins after the last time those digits were added.