The QCD axion is the leading solution to the strong-CP problem, a dark matter candidate, and a possible result of string theory compactifications. However, for axions produced before inflation, symmetry-breaking scales of f_a≳10¹² GeV (which are favored in string-theoretic axion models) are ruled out by cosmological constraints unless both the axion misalignment angle θ₀ and the inflationary Hubble scale H_I are extremely fine-tuned. We show that attempting to accommodate a high-f_a axion in inflationary cosmology leads to a fine-tuning problem that is worse than the strong-CP problem the axion was originally invented to solve. We also show that this problem remains unresolved by anthropic selection arguments commonly applied to the high-f_a axion scenario.