Calculating a little more carefully:
1 Planck time = 5.4e-44 seconds
Age of universe = 1.37e10 years * (365.242 days/year) * (24 hours/day) * (60 minutes/hour) * (60 seconds/minute) / (5.4e-44 Planck times/second) = 8.01e60 Planck times
log10(age of universe in Planck times) = 60.9
so log2(age of universe in Planck times) = log10(age of universe in Planck times) / log10(2) = 202.3
So we need 202-203 bits just to store current time stamps. But we should of course future-proof the design; perhaps we should invoke RFC 2550 dates (with particular attention paid to section 2.4.2)