Taking approximate $\mathrm{p}{K}_{\mathrm{a}}$ values of propyne 25, water 15.7, ethanol 16 and ammonia 35, which base converts propyne almost completely into its anion?

A base deprotonates an acid essentially completely only if the conjugate acid of the base is much weaker than the acid. $\mathrm{NH}{}_3$ ($\mathrm{p}{K}_{\mathrm{a}}$ 35) is far weaker than propyne (25), so amide does the job. Hydroxide, ethoxide and carbonate all have conjugate acids that are stronger than propyne.