For laundry:
* A load of laundry
* A basket of laundry
* A pile of laundry
For dishes:
* A stack of dishes
* A sink full of dishes
For other washing:
* A batch of washing
* A cycle of washing
You can also use more descriptive phrases like:
* A mountain of laundry
* A never-ending pile of dishes
* A thorough washing of the car
Ultimately, the best term to use depends on the specific context and what you want to emphasize.