A (generalized) Kempner series is a series of the reciprocals of numbers whose representations in a specified base omit a particular digit or set of digits. (The original Kempner series was the series $\frac11+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac1{10}+\ldots+\frac1{88}+\frac1{100}+\ldots$ of reciprocals of numbers without a 9 in their decimal representation.) All such series converge, by relatively straightforward comparisons with a geometric series.
The simplest case is the sum $\displaystyle\sum_{i=1}^\infty\frac1{2^i-1}$ of numbers whose base-two representation contains no zero. This sum is also known as the Erdős–Borwein constant and is actually known to be irrational (it shows up in analysis of certain binary search algorithms).
The next-simplest case is what might be called the Cantor-Kempner series of reciprocals of numbers whose base-3 representation contains no digit 2: $\frac11+\frac13+\frac14+\frac19+\frac1{10}+\frac1{12}+\frac1{13}+\frac1{27}+\ldots$ Computer investigation gives a value $\approx2.682853110966$ (and I believe all those digits are accurate), but looking this up in the Inverse Symbolic Calculator yields no results. Obviously the expectation is that this number is irrational (and even transcendental), but is there even a meaningful way of approaching such a sum?