It can take a LONG time to purge the air. If nothing else is wrong, that's
an IF. IF nothing else is wrong then just be patient. There are dangerous
ways to purge the air from the gas line but I wont mention them here. Be
careful.
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I have an overhead gas blower furnace in the garage. I had it
serviced last winter when the transformer fried.

Trying to fire it up for the first time this year, and the pilot won't
light.

I'm sure that the gas supply valve is open.

I thought maybe I just need to wait for the air to purge after opening
the valve, but I held the pilot button down for about 5 minutes, and
it still won't light.

Should I hear a hiss or smell gas, or the amount involved in a pilot
too small?