  Topological phases of matter have been extensively investigated in solid
state materials and classical wave systems with integer dimensions. However,
topological states in non-integer dimensions remain largely unexplored.
Fractals, being nearly the same at different scales, are one of the intriguing
complex geometries with non-integer dimensions. Here, we demonstrate acoustic
Sierpi\'nski fractal topological insulators with unconventional higher-order
topological phenomena via consistent theory and experiments. We discover
abundant topological edge and corner states emerging in our acoustic systems
due to the rich edge and corner boundaries inside the fractals. Interestingly,
the numbers of the edge and corner states scale the same as the bulk states
with the system size and the exponents coincide with the Hausdorff fractal
dimension of the Sierpi\'nski carpet. Furthermore, the emergent corner states
exhibit unconventional spectrum and wave patterns. Our study opens a pathway
toward topological states in fractal geometries.
