Is it possible to keep my stereo receiver powered on during ignition?

The red power wire on the back of the reciever is hooked up through the ignition. You need to have the it hooked directly to the battery in order to avoid interruption during starting.

Might have to be a large capacitor, and the risks of shorting out something with that instantaneous current capacity are not pleasant.

I'd wire at least one separate rechargeable battery (lead-acid automotive battery perhaps being cheapest and easiest to service, but Optima, AGM or NiMH are possibilities) that runs the stereo all the time, and arrange a line from the voltage regulator that keeps this battery charged properly as the alternator runs. The companies that supply high-current alternators for ambulances and similar vehicles will have external regulators that will work appropriately off your existing charging system. Have this turn on in the usual way, with 12V lead from the key, but put a time-delay circuit (and I'd think soft-off) for a reasonable delay. I'd also provide a button that would let you cycle 'hard off' when you wanted to cut power to the whole stereo rig, not just the head unit. Time delay would be related to safe discharge percentage of the battery you used.

This has the advantage that large power surges won't affect voltage elsewhere in the electrical system, and even if the battery deep discharges, it will not drag your starting/running battery down or compromise its lifetime.

I'm somewhat nervous about using modern Li-ion for this, as the energy density for shorts or in accidents is getting high enough to pose a chemical fire danger.