Help on (non-working) LM317 constant current circuit

I have used red LEDs with MegOhm+ series current limiters as pilot lights in 110Vac circuits (more for continuity checking than pilots, but you get the picture. Voltage isn't really the issue.

Just wanted to add, you can do that only if you either have two LEDs connected anti-parallel (parallel but pointing different directions) or one LED with a regular diode connected to it in anti-parallel. That is because LEDs generally don't survive even modest current levels when put in reverse breakdown, and the reverse breakdown voltage is not very high.