Technically there are an infinite number of these, but they aren't that easy to find. An interesting one is multiplying a series of 1's with itself, for example: 1*1=1, and 11*11 = 121. This scales up to nine 1's:

111111111

*

__111111111__
= 12345678987654321

I call this 'elegant' as not only is it a palindrome, but it is 'in-order', first counting up, then counting down. Trying ten 1's also results in a palindrome, however two zeros get introduced between two 9's which make it a little less appealing to the eye, and arguably inelegant (12345678900987654321). Trying eleven 1's doesn't result in a palindrome because humans use the base ten numeral system (a positional numeral system with a radix of ten), most likely because we have ten fingers and ten toes. (Also the reason that in the English language 'digit' and its translation in many languages refers to both a mathematically expressed numeral and the anatomical name for fingers and toes). Eleven 1's multiplied by eleven 1's results in the inelegant, non-palindrome 123456790120987654321.

On a related note, many cultures did not use the base ten system such as the Maya who used base twenty, using the sum of all fingers and toes. Others have used octal systems, counting the spaces between fingers and toes. Computers prefer the binary system because they have only two fingers, one on each hand.