Which of the following is least basic?

The basicity of an amine depends on the availability of the lone pair of electrons on the nitrogen atom. When the lone pair is delocalised (especially into electron-withdrawing groups like carbonyl groups), the basicity decreases significantly.

Consider each compound: $$(C_2H_5)_3N$$ and $$(C_2H_5)_2NH$$ are simple aliphatic amines with alkyl groups that donate electron density to nitrogen through the inductive effect, making them reasonably basic.

$$(CH_3CO)NHC_2H_5$$ is an amide (N-ethylacetamide) where the nitrogen lone pair is delocalised into one carbonyl group through resonance: $$-N-C=O \leftrightarrow -N^+=C-O^-$$. This significantly reduces the basicity compared to simple amines.

$$(CH_3CO)_2NH$$ is a diacylamide (diacetamide) where the nitrogen lone pair is delocalised into two carbonyl groups. The presence of two electron-withdrawing acetyl groups causes even greater delocalisation of the lone pair compared to a mono-acylamide, making this compound the least basic of all the given options.

Therefore, $$(CH_3CO)_2NH$$ is the least basic compound among the given choices.