Given this polygon:
I'm trying to select the inner one, but can't find a simple way using either selection by location or selection by expression
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Use Select by expression with this expression and change the size of the buffer (end of line 2 - here: 0.0001) depending on the size of your polygons/the CRS you use (in most cases, it should work without adaptions).
Explanation: the expression evaluates if the outer boundary of a buffer around a polygon feature is within the nearest neighboring polygon.
within ( boundary (buffer( $geometry,0.0001)), array_first (overlay_nearest (@layer, $geometry)) )