I have in the past dispensed with a box spring and used 1/2" plywood to almost completely cover the mattress base. The advantage is that the mattress has uniform support and never shows the dents from the slats.
The main disadvantage is that 1/2" is rather heavy for taking off on a weekly basis to vacuum under the bed.
Would I get away with 3/8", or even with 1/4", plywood? My hunch is that 1/4" is plenty of thickness to support an average-weight (150-200 lb.) adult. But it has to also support the weight of the adult standing, for otherwise a simple exercise like changing the ceiling lamps would crack the base.
Rephrasing and summarizing: what is the fir/spruce plywood thickness needed to support an individual on a span of 38" (half the span of a King-sized bed)?