A lot of NES mappers are not much help in the "special effects" department, they just allow you to page the ROM in order to access more of it. The functions that are actually helpful in this area are CHR-ROM bankswitching (in small chunks, the 4K of the MMC1 is not really good for effects) and scanline/cycle counters, both present on the MMC3, which is one of the most common mappers, but AxROm and UxROM boards are also very common, and those don't do anything besides bankswitching PRG-ROM.
Battletoads is a good example of a game that uses such a simple mapper and makes incredibly good use of the capabilities of the system. You won't find many good examples in NROM games, but that's just because every developer soon started using boards with more PRG-ROM size. Games started to get larger after the NES was released, everyone wanted to play games with more levels, more graphics, more songs, that weren't so repetitive.
In
this post, for example, I linked to a video of an effect in Bucky O'Hare that I find very impressive, and can be done without a mapper.
As for the Genesis, I think very few games have any special chips, most just used the normal capabilities of the console. The Genesis can access 4MB or ROM, so only games larger than that (not many I think) would need bankswitching.