The voltage of a brand new AA alkaline cell is about 1.5V, barely enough to light even red LEDs. After less than 10% of its energy is used, its voltage is about 1.2V, which isn't usually enough. So the almost new cell won't light an LED. Of course, a 1.25 NiMH or NiCd is hopeless even when fully charged.
Joule Thief doesn't create energy, it just takes that 90% of the battery which lies below 1.5V and makes it useful to an LED. By the time the cell is down to 0.8V most of the energy has been harvested.
Anything that causes a halt to flux increase will end the feedback and permit oscillation. Core saturation works, but some of the transformers I see won't saturate. In those cases, the tiny transistor runs out of beta at fairly low collector currents.