Take any number and keep appending 1's to the right of it. Are there an infinite number of primes in this sequence?

I think this is an open question. Lenny Jones gave a talk in which he noted that the numbers 12, 121, 1211, 12111, 121111, etc., are all composite - until you get to the one with 138 digits, that's a prime.

Jones' work appears in the paper, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite integers?, Amer. Math Monthly 118 (Feb. 2011) 153-160. He finds that 37 is the smallest positive integer such that you get nothing but composites by appending any positive number of ones. It seems to be easier to find a sequence with no primes than a sequence which you can prove has infinitely many.

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