  We study the classic single-choice prophet inequality problem through a
resource augmentation lens. Our goal is to bound the
$(1-\varepsilon)$-competition complexity of different types of online
algorithms. This metric asks for the smallest $k$ such that the expected value
of the online algorithm on $k$ copies of the original instance, is at least a
$(1-\varepsilon)$-approximation to the expected offline optimum on a single
copy.
  We show that block threshold algorithms, which set one threshold per copy,
are optimal and give a tight bound of $k = \Theta(\log \log 1/\varepsilon)$.
This shows that block threshold algorithms approach the offline optimum
doubly-exponentially fast. For single threshold algorithms, we give a tight
bound of $k = \Theta(\log 1/\varepsilon)$, establishing an exponential gap
between block threshold algorithms and single threshold algorithms.
  Our model and results pave the way for exploring resource-augmented prophet
inequalities in combinatorial settings. In line with this, we present
preliminary findings for bipartite matching with one-sided vertex arrivals, as
well as in XOS combinatorial auctions. Our results have a natural competition
complexity interpretation in mechanism design and pricing applications.
