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I'd say ask an engineer...
40mm solid beech is not going to flex much over 300mm, but the units only really support at the sides, right? or is there a rail at the back against the wall? I feel like fixing down to a rail against the wall would make a difference, also if you could add a rail along the top/front of the units but might not be space for that.
Is it block laminated? There could be risk of it splitting I guess, depending how much you load the overhang. 300mm doesn't seem that controversial though.
hoefla-
I'd say ask an engineer...
Ha! I should probably get my calculator out. I reckon it rather depends on how heavy my cereal bowl will be...
Here's a rough sketch. I've got opportunities to add support/ legs later, so in some ways I'm just thinking out loud. It's Worktop Express beech, so staves glued together. Their assembly instructions suggest overhangs exceeding 200mm need a supporting leg, but then their documentation for those legs says they're needed for overhangs exceeding 300mm! Perhaps the confusion is between worktop thicknesses of 27mm and 40mm.
Cupcakes
Breakfast bar overhang of 300mm in 40mm thick solid beech worktop. Will I get away with this or should I add a supporting leg? (The overhang juts out from a 600mm deep base unit).