This question is far from having a real answer, in the last century with the dawn of the computer-era, estimations arose ranging from 10^9 to 10^20 bits, based on "hardware" considerations. These are based on an estimation of about 10^13 synapses in a human brain each corresponding to a few bits. More empirical studies tend to lower this number to about 10^8-10^10.
Some theories, based on quantum mechanics, lead to totally different numbers, many orders of magnitude above 10^20, but those take into consideration the entire amount of "information" involved in the process we call "consciousness". and involve calculations that simulate virtual black-holes forming inside your brain...
Perhaps a more interesting question would be to ask "how much can we recall" (upon request or perhaps in random order). I could write about this forever, instead - I will leave you with a different question, which hopefully might help you understand the complexity of the matter.
Can a human generate a very long random string? (say, 10000 bits) given only a notepad and a pencil...
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see also:
http://www.merkle.com/humanMemory.html
http://www.merkle.com/brainLimits.html
http://xxx.lanl.gov/PS_cache/quant-ph/pdf/9510/9510003.pdf