If we can delete the digits from N one at a time from the right and still get a prime, then N is a right truncatable prime. If we can delete the digits from N one at a time from the left and still get a prime, then N is a left truncatable prime. Are there any primes in which we can repeatedly delete any digit and still get a prime at each step? If so, each digit would have to be prime, and no digit could occur twice, so this would be a short list: 2, 3, 5, 7, 23, 37, 53 and 73.
To make the search more interesting, a deletable prime has been defined ([Caldwell87]) to be a prime that you can delete the digits one at a time in some order and get a prime at each step. One example is 410256793, because the following are (deletable) primes: