It depends on the minimum size of the what you will consider a hole. Check out the video from VSauce on YouTube about how many holes a human has. [this video]
Explores how many holes a human has depending on what counts as a hole
Vsauce has an interesting video regarding topology[0]. Probably a bit rudimentary for anyone already familiar with the math, but still entertaining in the typical vsauce way. [0] [this video]
Entertaining, somewhat rudimentary introduction to topology
This vsauce video is about humans, but I imagine pretty much every mammal will be just about topologically equivalent. As it turns out we humans have quite a few more holes than fleshy donuts would. [this video]
Shows humans have more holes than a donut topologically
The Vsauce video in sibling comment at 12:59 shows an animation from the two-rings to the T-shaped pipe, or even a better animation at 15:59. 12:59 [this video] 15:59 [this video]
Animations of rings morphing into a T-shaped pipe at given timestamps
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