H_n = 1 + 1/2 + ... + 1/n ≈ ln n + 0.577.
It diverges, which surprises people, but so slowly that reaching a sum of 100 requires more terms than there are atoms in any practical sense.
Where it appears in interviews. The coupon collector problem: collecting all n coupons takes n H_n draws, roughly n ln n. Also the jeep-in-the-desert fuel problem, and the expected number of records in a random sequence.
Whenever an answer involves "the last one takes as long as all the rest", a harmonic sum is usually behind it.