Electronic – Two LM317 Based Voltage and Constant Current Regulator Circuit

note: question was originally about a current limiter

The LM317 with the single series resistor between output and adjust input is actually a fixed current source, not a current limiter. You don't need the LM317 to create a current limiter, a few discrete components will do:

For a limiting at 2mA you select a 330\$\Omega\$ resistor for \$R_{SENSE}\$. If there flows 2mA through it Q2 will start to conduct and reduce the base voltage of Q1, so that its current is cut off.

edit (re changed question)
Maybe you're focusing too much on the LM317. If you need a constant current you could use the LM234 which is a programmable current source for up to 10mA. You set the current with a resistor.

If you had a short in place of the potentiometer, you should have read 1.25 volts (or so) at the output of the LM317, not 12.5 volts.

When you changed your meter to read current, presumably moving the red lead to the "10 Amp" socket, the meter would have very low resistance. When you then connected the meter between the LM317 output and ground, a very large current would flow, apparently causing the LM317 to explode.

If I recall correctly, the LM317 is spec'd to deliver 1 amp, with due regard to voltage drop, power dissipation, heatsinks and phase of the moon - that only states the current that it can safely carry under suitable conditions - it DOES NOT limit the current to that value!

The current measurement function of a multimeter should only be used with the meter connected in series with a load whose current drain you wish to measure. It must NEVER be connected directly across a power supply.