IPv4 provides four billion IP addresses.
Mathematically, IPv6 will provide
340,282,366,920,938,463,463,374,607,431,768,211,456 IP Addresses.
Following is an explanation of the number:
340 - undecillion
282 - decillion
366 - nonillion
920 - octillion
938 - septillion
463 - sextillion
463 - quintillion
374 - quadrillion
607 - trillion
431 - billion
768 - million
211 - thousand
So in all words, that would look like:
Three hundred and forty undecillion, two hundred and eighty-two decillion, three hundred and sixty-six nonillion, nine hundred and twenty octillion, nine hundred and thirty-eight septillion, four hundred and sixty-three sextillion, four hundred and sixty-three quintillion, three hundred and seventy-four quadrillion, six hundred and seven trillion, four hundred and thirty-one billion, seven hundred and sixty-eight million, two hundred and eleven thousand, four hundred and fifty-six.
That is one big number.
IPv6 is more complicated but we should be good on IP Addresses until we start mingling with other Galaxies.