Which of the following alkyl bromides undergoes E1 elimination (and ionisation) most readily?

The rate of E1 is governed by the stability of the carbocation formed in the slow step. The tertiary cation $(\mathrm{CH}{}_3){}_3\mathrm{C}{}^{+}$ is stabilised by nine C-H hyperconjugating bonds and three methyl groups, so the tertiary bromide ionises fastest.