Depending on your box, you may be able to buy a TWIN breaker such as this one:
Circuit Breakers - Square D - QOT1520
This will place two independent circuits in a single space. Note you can get them in the same amp rating or different ones.
Somewhere on your panel, it has a "Maximum number of poles" rating. This number cannot be exceeded. Basically, it's the number of wires attached to breakers
This would not replace your dryer breaker, but an existing 120V circuit. You'll then have 2 circuits in one space.