Energy-efficient LED driver circuit
TL;DR version:

Use two alkaline AA cells, a 2.7 kOhm resistor and the LED in series, such that the LED is provided the bare minimum current that lights it up sufficiently.



Summarizing from chat discussion around this question. 


Determine empirically what minimum current through your specific LED provides sufficient light to be acceptable as "staying on". Done: 0.25 mA
Measure precisely the forward voltage of the LED at around twice that current. Done: 1.86 Volts at 0.5 mA

Why the higher current? So that the "staying on" condition remains valid even when the current drops to half the starting value due to battery depletion.

Determine nominal voltage for your power source: Alkaline AA = ~ 1.5 x 2 = 3 Volts
Simply add a series resistor to the LED such that ~ 0.5 mA is supplied to the LED.  Calculation: R = (Vbatt - Vled) / Iled = (3 - 1.86) / 0.0005 = 2.28 kOhms
Next higher E12 value = 2.7 kOhms, gives 0.42 mA, well within our desired current range, so we are done.


Now, calculating power consumed and longevity of the solution


Circuit power consumption: P = V x I = 3 x 0.00042 = 1.27 milliwatts
Capacity of batteries: 

Assuming Energizer E91 Alkaline AA. 
From Milliamp Hours Capacity chart, extrapolating to discharge rate less than 1 mA, but discharging only to 1.27 Volts gives approximately 3000 mAh. This assumption is such that the voltage does not drop below 2.535 Volts for 2 batteries in series, so that current across LED does not drop below 0.25 mA.
Keeping that as a worst-case number, 2 cells = 6000 mAh (Approximation!)
At 0.42 mA, minimum longevity = 6000 / 0.42 = 14285.7 hours = ~ 595 days, close to 2 years.

Even if we discount the calculations above significantly, this solution will still give around a year of battery life with a good estimation buffer. 




As per experiments conducted and mentioned in chat, using a resistor of 2.4K, it consumes 0.25mA at 2.5V, 0.42mA at 3.0V. This proves that the calculations above are conservative, a longer duration is not inconceivable.