I think Black Copper Marans are most likely to produce the darkest brown eggs, but (as has already been advised) the exceptionally dark egg layers are difficult to find...and costly. I've been breeding them for several years, seeking that elusive "super dark" egg - and friendly birds - and was fortunate to find new stock locally that met both criteria. And this isn't even the darkest shade: some are nearly espresso-colored or have actual black spots. It's so amazing to find these in nest boxes, nestled among the pink/green/blue shades! 
It's highly unlikely you'd get eggs like this from hatchery stock, though. If you really want very dark eggs, buy from a reputable breeder and be prepared to pay for that level of quality. Best of luck in your quest!

It's highly unlikely you'd get eggs like this from hatchery stock, though. If you really want very dark eggs, buy from a reputable breeder and be prepared to pay for that level of quality. Best of luck in your quest!