Is this modified coffee cup equivalent to some n-fold torus? 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.
 A: More precisely, we are talking abiout the surface of the mug. Without this extra, this would be a simple torus. If the tube were just a filled bar, that would make one extra handle. "Drilling" a hole through that bar adds another handle. So what we have is a three-fold torus (three donuts smashed together). It's a fun mental exercise to deform the mug to "separate" the handles.
A: It is a surface of genus 3.
First disregard the outer handle - we can always just add that back on later. We then have to show that the middle bit has genus 2.
To see this, we flatten the cup out, so it looks like a pancake with a thick handle. I'll show how we can go from a pancake surface (a flat sphere) to our surface by adding two handles. 
First, we add one handle going over the top, not cutting through the pancake shape. If you made this from clay, it would have a solid interior. Next, you drill out the inside of this solid handle, which is the same as adding a second handle going through the first. This process has added two handles, so counting the original one we removed, we see that the surface has genus 3.

