AHHH!
units, the devil is in the details.
watts is a rate!
"The laptop consumes 15 - 20 watts. I bought a panel rated for 18 watts and assuming 3.5 hours of full sun, this would generate somewhere around 60 watts, about enough to run the laptop for three hours."
your laptop does not consume watts. think of watts like miles per hour. it is a rate. your laptop consumes at a rate of 20 watts. running for 3 hours it would consume 60 watt-hours.an 18w module produces at a rate of 18w at standard testing conditions, which is about 25% better than the real world. if you bought a pv module rated at 18w and your site was located in imaginery STC land , you could have 63 watt-hours energy production in one day.
now more math:
60 watt-hours per day load.
x2 = 120 watt hours of storage required for 1 day
x3 = 360 watt hours of storage for 3 days of load
/12v your assumed system voltage = 30amp-hours
which would be a battery like this.
http://store.altenergystore.com/Batteries/Batteries-Sealed-Agm/Universal-Ub12350-12V-34Ah-20Hr-Sealed-Agm-Batt/p1999/i would not keep the battery outside if at all possible. certaily in some sort of enclosure, even if not fancy battery box.
james Alt-E staff