We demonstrate that stacking topologically trivial layers, under enforced symmetry restrictions, yields emergent topological phases with protected boundary states. Remarkably, the number of layers itself acts as a topological switch, enabling the system to host topological bound states in the continuum (BICs). Analytically, we reveal that an odd-layer configuration not only renders the spectrum gapless but also supports BICs protected by interlayer reflection symmetry. Combined with entanglement-spectrum calculations, this confirms that odd-layer systems indeed support topological BICs. We provide experimental confirmation of these topological states in stacked acoustic lattices. Our findings establish a previously overlooked pathway to topology and demonstrate a readily applicable strategy for realizing exotic states in a wide range of artificial material systems.