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I came up with this line somewhere:

To properly experience $n$ dimensions the retina of the organism should be at least of $n-1$ dimensions.

And human eye has $2$ dimensional retina, so we can experience at most 3 dimensions properly.

Now my question is:

  1. Do more than 3 dimensions exist?

  2. What is the proof that 'there exist only 3' or 'more dimensions do exist'?

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Well, I suppose we could add time as a 4th. However, apart from that there is no experimental evidence whatsoever for more than 3 spacial dimensions. String Theory and its friends remain theories with negligible experimental evidence in their favor.