WhatsApp says that there were 10 billion incoming messages and 17 billion outgoing messages processed in one 24 hour period. That makes for 27 billion total messages, which beats the old record of 18 billion, which was set on New Years Eve 2012. The incoming and outgoing message totals are not even, according to WhatsApp, because the outgoing message total gets inflated by group message.
It's hard to say how the number stacks up to other messaging platforms, because few others really give out those numbers. We had heard recently that BBM is processing about 10 billion messages a day by its 60 million users. WhatsApp recently said that it has 200 million active monthly users, but has not said how many total registered users are on the platform.