2
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
3
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.
Thanks, can you give a reference on the limit per output? – morsecoder – 2015-01-28T22:33:25.800
@StephenM347 It's not enforced anywhere in the code - but the value field in outputs is an int64_t. There is also a separate check that the outputs are less than MAX_MONEY, but that can be changed pretty easily.
2This is a minor point of terminology, but the Dogecoin equivalent of Satoshi are called Koinu (romanized Japanese word for puppy). – user1205901 - Reinstate Monica – 2015-07-13T11:49:21.327