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So, I just was trying to get back into blender, and I made a cup. Unfortunately, this cup has some issues right at the bottom of the inside, and I can't figure out why.

You can download the blend file here. And just for good measure, here is a screenshot of the problem.

cup with problem

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That is caused from the triangle fan at the bottom of your cup, and the subdivision surface modifier.

To fix it delete the vertices at the center of the bottom of the cup.
Select the edge loop along the bottom, press E to extrude then S to scale it smaller. Repeat that three times, and for the last time scale to 0. Then remove doubles.

This is how it will look after extruding and scaling three times. This will give acceptable results when using a subdivision surface modifier.
good topology

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