Robin's theorem states that the Riemann hypothesis is equivalent to the inequality σ(n) < e^γ n log log n for all n > 5040, where σ(n) is the sum of divisors of n and γ is Euler's constant. It is natural to seek the first integer, if it exists, that violates this inequality. We introduce the sequence of extremely abundant numbers, a subsequence of superabundant numbers, where one might look for this first violating integer. The Riemann hypothesis is true if and only if there are infinitely many extremely abundant numbers. These numbers have some connection to the colossally abundant numbers. We show the fragility of the Riemann hypothesis with respect to the terms of some supersets of extremely abundant numbers.