Driving LED with 555 timer and BC639 transistor
The value of the base resistor has nothing to do with the range of the PWM. 

R3 should be more like 10K to fully saturate the transistor (forced beta in the range of 20). 

When the output is high (LED 'on'), the capacitor has to charge through R2, which means that the "on" time is limited by the time constant of R2 and C1. 

You can try "flipping" the output (if you have an ancient design bipolar 555 you may need a pullup to turn the transistor off fully), so use a PNP transistor and the same value of base resistor. Then reverse the CCW and CW sides of your pot. 

That will mean your LED output won't go as close to 100%, but that will have no visual effect-- the slight amount 'on' is much more noticeable than a slight reduction in full brightness. 

The output will still not be completely off, but it will be greatly reduced in brightness. A small capacitor, at a guess around 1nF between base and emitter of the transistor can eliminate that short turn on, at the expense of the transistor running a bit warmer.