The US Marine Corps Band ("The President's Own") wear unique "inverted" uniforms, where the tunic is red, not blue with red piping as normal marines have it. Officers and senior enlisted have more ornate and very historical uniforms.

As for other services, the US Navy Sailors on the USS Constitution wear period Navy uniforms. That's the only other example I can think of other than most SOF units being allowed to customize and open purchase much of their kit and grow facial hair.

The Service Academies also each have a very traditional uniform with a high neck and double row of buttons down the front. (full dress blue/gray)

Finally, Officers of the rank O-4 and above and enlisted E-7 and above have dinner/mess dress blues/whites, which are extremely formal and traditional and vary by service.