The familiar joke is that a coffee mug is topologically equivalent to a donut.
I own a coffee mug that is essentially a regular coffee mug with a tube going the middle. I'm not sure how to describe it, but this youtube video shows someone handling a coffee mug just like it. https://www.youtube.com/watch?v=n-SsPJXibXs
My friends and I cannot decide whether or not this is homeomorphic to a 2-fold torus or a 3-fold torus or even some other shape, and I lack the math skills to figure it out myself.