In the gas phase, which of the following is the strongest base?

With no solvent, only the electronic effect matters. Each methyl group donates electrons and also polarises to stabilise the ammonium ion, so basicity rises as $\mathrm{NH}{}_3<\mathrm{CH}{}_3\mathrm{NH}{}_2<(\mathrm{CH}{}_3){}_2\mathrm{NH}<(\mathrm{CH}{}_3){}_3\mathrm{N}$. The tertiary amine is strongest in the gas phase.