Step 1: Test each option:
Methanamine, propan-2-amine and ethanamine each carry an $-\text{NH}_2$ group, so they all give the carbylamine reaction.
Step 2: Odd one out:
Triethylamine has three alkyl groups on nitrogen and no N-H, so it is the one that does not react.
Final Answer:
The answer is $N,N$-diethylethanamine, option (C).
\[ \boxed{N,N\text{-Diethylethanamine}} \]