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I have a PostGIS database with a very large amount of geometries (circles), most of them overlapping. I'm trying to come up with a way to find the nearest point (from your current location) that is not within any of these circles.

Is this possible with a direct query?

So far I thought of a fishnet approach of 1x1 km cells that check for overlap with any of these circles. This works, but is not very accurate.

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Since you are using a PostGIS database, you could use the st_closestpoint function. One argument would be your current location, the other one your inverted circle geometries.

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