Many

This is perplexing but fascinating:

If observers are an integral part of the cosmic formula, then it may not matter how many universes exist - just how many a single observer can tell apart. If the observer is a person, that depends on how many bits of information the brain can process. "Based on the number of synapses in a typical brain, a human observer can register 10 ^16," says Linde. That means humans can differentiate 10^10^16 universes, which is much more manageable than the 10^10^10,000,000 Linde and Vanchurin found to start with.

See also a report on Quantum to Cosmos.