A repunit is a number using only one digit. The numbers 11 and 1111111111111111111 are examples of repunit primes. The number of ones in such a repunit primes can be 2, 19, 23, 317, 1031, 49081, 86453, or 109297.
An almost repunit number repeats a digit but one digit is different, for example, 222222922. The primes of this form make interesting patterns, as shown in this Demonstration. Whether there is any pattern to these is an unsolved problem.