Which of the following alkyl bromides reacts fastest with water by an SN1 pathway?

The rate of SN1 depends on the stability of the carbocation formed. Only the tertiary bromide gives a tertiary carbocation; the primary and methyl bromides would give very unstable cations. So $(\mathrm{CH}{}_3){}_3\mathrm{C}-\mathrm{Br}$ reacts fastest by SN1.