To all the people saying "it's not possible because you have more possible paths than integers to store them in": no. The poster never specified an implementation language; some languages support arbitrary-length integers. Python, for example.
Say we take the 32,000 character paths as the limit mentioned in one of the other comments. If we have 256 different characters to use with paths we get:
Python 2.5.1 (r251:54863, May 18 2007, 16:56:43)
[GCC 3.4.4 (cygming special, gdc 0.12, using dmd 0.125)] on cygwin
Type "help", "copyright", "credits" or "license" for more information.
>>> 32000L**256L
20815864389328798163850480654728171077230524494533409610638224700807216119346720596024478883464648369684843227908562015582767132496646929816279813211354641525848259018778440691546366699323167100945918841095379622423387354295096957733925002768876520583464697770622321657076833170056511209332449663781837603694136444406281042053396870977465916057756101739472373801429441421111406337458176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000L
>>>
Notice how Python represents that just fine? Yes, there's probably a better way to do it, but that doesn't mean it's impossible.
EDIT: rjack pointed out that it's actually 256^32000, not the other way around. Python still handles it just fine. The performance may leave something to be desired, but saying it's mathematically impossible is wrong.