Step 1: N-methylpiperidine is a cyclic tertiary amine, a six-membered ring containing one nitrogen already bearing a methyl group.
Step 2: Treating it with excess methyl iodide converts the ring nitrogen into a quaternary ammonium iodide.
Step 3: Moist silver oxide exchanges iodide for hydroxide, and heating causes Hofmann elimination.
Step 4: Because nitrogen is part of a ring, this first elimination opens the ring into an open-chain amino-alkene, \((CH_3)_2N-CH_2-CH_2-CH_2-CH=CH_2\).
Step 5: This open-chain amine is again methylated, treated with \(Ag_2O\), and heated a second time, removing trimethylamine and forming a second double bond.
Step 6: The double bonds end up at carbons 1 and 4 of the five-carbon chain, giving \(CH_2=CH-CH_2-CH=CH_2\), penta-1,4-diene.
\[\boxed{\text{Penta-1,4-diene}}\]