While working with Brooke Clarke's JouleThief bulb replacement project (Ref 1), I designed a regulated LED driver operating from a single 1.5V cell.
Current regulation provides constant light output over the life of the battery.
Please note R5 could be omitted and replaced with a jumper.
Here is the circuit simulation under Linear Technology's SwitcherCAD (Ref 2):
This shows the current in R2 at supply voltages of 0.8, 1.0 and 1.5V.
At high battery voltage, stability is not great (see the small oscillation as the current goes into regulation, on the larger trace, between 1.7 and 2.5 mS).
Here is a link to the schematic in SwitcherCAD format: JouleThief2.asc