Step 1: Recall the oxidation rule.
Alkaline $KMnO_4$ oxidises a side chain on a benzene ring down to a $-COOH$ group, but only if the carbon attached to the ring (the benzylic carbon) carries at least one hydrogen. No benzylic H means no oxidation.
Step 2: Check 1-phenylpropane.
Here the ring is attached to a $-CH_2-$ group, which has benzylic hydrogens, so it oxidises to benzoic acid.
Step 3: Check 2-phenylpropane.
The ring is attached to a $-CH(CH_3)_2$ carbon, which still has one benzylic hydrogen, so it also gives benzoic acid.
Step 4: Check acetophenone.
Acetophenone is $Ph-CO-CH_3$; its side chain is oxidised away to leave benzoic acid as well.
Step 5: Check 2-methyl-2-phenylpropane.
This is tert-butylbenzene, $Ph-C(CH_3)_3$. The benzylic carbon is bonded to three methyl groups and the ring, with NO hydrogen on it. Without a benzylic H, it resists oxidation and does not give benzoic acid.
Step 6: Conclude.
The compound that does not form benzoic acid is 2-methyl-2-phenylpropane.
\[ \boxed{\text{2-methyl-2-phenylpropane}} \]