Take any number and keep appending 1's to the right of it. Are there an infinite number of primes in this sequence? Ignoring sequences that are always factorable such as starting with 11, Can we take any other number such as 42 and continually append 1s (forming the sequence {42, 421, 4211, ...}) to get a sequence that has an infinite number of primes in it?

Unless prevented by congruence restrictions, a sequence that grows exponentially, such as Mersenne primes or repunits or this variant on repunits, is predicted to have about $c \log(n)$ primes among its first $n$ terms according to "probability" arguments.  Proving this prediction for any particular sequence is usually an unsolved problem. 
There is more literature (and more algebraic structure) available for the Mersenne case but the principle is the same for other sequences.
http://primes.utm.edu/mersenne/heuristic.html
Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The New Mersenne Conjecture." Amer. Math. Monthly 96, 125-128, 1989
