There's a very old (at least as old as ancient Rome) stock character called the tricky or clever slave. This is a character in a menial position who, by exercise of wits and determination, is actually running things. To be impressive, the character must be much less powerful than his or her boss. In turn, the boss usually is ill-suited to power out of pure stupidity, or some form of upper - class helplessness. The character will often bring about a happy ending for both him- or herself and the boss. It seems like these characters have been heavily concentrated as domestics and in the military:

Lt. Commander McHale, McHale's Navy
Sergeant Bilko, Phil Silvers Show
Geoffrey, Fresh Prince of Bel Air
Benson, Soap/Benson
Hazel, Hazel
Colonel Hogan, Hogan's Heroes
Sergeant O'Rourke, F Troop
Rochester, Jack Benny Show
UK: Blackadder, esp. Goes Forth (fourth season)

Sometimes family members have some of the same "operator" characteristics: Danny Partridge, Patty Lane, Gracie Allen. Some fantasy characters, too, like Mr. Ed or Top Cat.

What are some other characters of this type? Are there any that aren't domestics or in the military?

I'd say early Radar in M*A*S*H qualifies.