I'm new to electronics and wiring, and wonder if a 15 ohm resistor will drain a 1.5v battery (specifically a Duracell Alkaline AA) in a reasonable amount of time to a super-intense white LED.
Does anyone know if this will work?
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Assuming you mean a 1.5V battery with a 15\$\Omega\$ resistor in series with a white LED, no it will not drain the battery very fast at all. You will get something approaching the shelf life of the battery (could be years).
Unfortunately, the LED will also remain quite dark, as they require about 3V to operate.
Presumably your proposed schematic looks like this:
If you build the circuit and measure the voltage across the resistor you can use Ohm's Law to calculate the current drawn. Or if you have a V-I plot for the diode you could use a load line analysis to calculate the current drawn analytically. Either way, you can use the current drawn to calculate the shelf life using Duracell's datasheet for its AA batteries (which can be found here). It includes plots for service life based on constant current consumption.
Unfortunately, as @Spehro Pefhany has pointed out, a white LED usually requires a voltage drop in the 3V range so the current consumption will be very low and the battery will last a long time. You could stack two or three 1.5V batteries to get around this problem. Since the batteries are in series the same current would be drawn from all of them.