The rec/reg is the rectifier/voltage regulator in a single unit. It is responsible for converting AC voltage from the alternator (stator) into DC to charge the battery and regulate that voltage to 14.8v so the battery does not overcharge. There are only a couple of modules along with the wires that will go up in smoke. Another unit is the CDI unit. If the CDI burns up you will not have spark and the engine will not run.
If you run without a battery, the voltage regulator has no reference voltage and the voltage is unregulated. Without a battery, the lights will burn out and other DC operated accessories can burn up also as the unregulated DC voltage may be as high as 28 volts. When the rec/reg is not operating as designed, the unregulated voltage may burn out the diodes that the rectifier is comprised of which in turn melt the potting material (epoxy) that encapsulates the components to protect them from the elements and aids in cooling them also.
When the diodes overheat, the epoxy melts and makes smoke indicating a problem. If the diode (a semiconductor) shorts out instead of opening like a fuse and a battery is installed, the battery discharges through the shorted diode again overheating it usually until the short opens and the current flow stops.
Rule # 1 - don't operate an vehicle designed for a battery without the battery or with a defective battery.
Good luck in finding where the smoke came from and which module failed.