Arbitrarily large Morita Frobenius numbers

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large $\mathcal{O}$-Morita Frobenius numbers are constructed.