[math-fun] Car Talk and prime numbers

Dear Funsters, The radio program Car Talk today was discussing the problem of finding the longest English word which has the property that you can repeatedly delete one letter and still have an English word, all the way until the empty word is left (the best solution mentioned had 11 letters). At each step you can choose which letter to delete. This led me to wonder about the analogous question for primes. This must be well-studied. What's the largest prime known with property that you can repeatedly delete a digit (you get to choose which digit) and still have a prime, all the way down to the empty word? I don't want to sign this.