BanachTarski Paradox 

If you cut a sphere into a finite number of pieces and reassemble the pieces, will the resulting shape always have the same volume as the original sphere? NO !!!
The Banach  Tarski Theorem states that it is possible to dissect a ball into finitely many pieces (in fact five will do) which can be reassembled by rigid motions to form two balls each with the same size as the original !!!
To give a simplistic insight in how you can end up with more than you started
... Imagine a perfect dictionary (without definitions !!!) containing every possible word (permutation of letters) however long. It would contain a countably infinite number of entries. (see countable sets).
It would contain:
AACAT  BACAT  ...  ZACAT 
ACAT  BCAT  ...  ZCAT 
ADOG  BDOG  ...  ZDOG 
AELEPHANT  BELEPHANT  ...  ZELEPHANT 
etc.  etc.  etc.  etc. 
This dictionary could then be cut into 26 identical perfect dictionaries when the first letter of every entry in the dictionary has been ignored.
http://mathworld.wolfram.com/BanachTarskiParadox.html
http://en.wikipedia.org/wiki/BanachTarski_Paradox