How many Dogecoin-satoshis will ever exist? In Bitcoin, there will only ever be ~21,000,000 coins (slightly less, technically, but I'll just use 21 million for easy calculation). Each bitcoin is 10^8 satoshis. This means there be a maximum:

2,100,000,000,000,000 satoshis



What is the equivalent number for Dogecoin? 
Is it larger than the maximum 64 bit unsigned integer?
Dogecoin has no maximum, because it mints 10k dogecoins per block, forever.

But to answer the spirit of your question, there are 100 million indivisible parts to a dogecoin. You can see that here.

static const int64_t COIN = 100000000;


There are currently 97 billion dogecoins. This is a bit more than 2^63 satoshis.


  Is it larger than the maximum 64 bit unsigned integer?


No, but it wouldn't matter if it was. There isn't a limit of 2^64 satoshis on the total money in Bitcoin (or altcoins). However, there is a limit of 2^64 satoshis per output.