A typical pun is a pun because it is a play on words. Usually this is done by using two seperate meanings of a word.
Has there ever been a pun that has had 3 or more ‘plays’ in the same instance?
I take it you’ve never heard Elvis Costello? 
I personally have written lyrics with 4 or more meanings to the words, of course some are more symbolic or mere word play than straight up puns, but 3 or more is definitely possible.
It ought to be easy enough to do with “set” which has about as many separate meanings (both as noun and as verb) as anything in the dictionary. At one time, maybe still, it had the most meaning of any English word.
Yes, “set” is fun to look up on the visual thesaurus.
A sign saying “PARK POLICE SNOWMOBILES HERE ONLY” seems to have at least 8 possible meanings, because there are 4 words that “ONLY” might be modifying, and because “PARK” can mean “leave it” or “public park”. Perhaps there are 9 meanings because “ONLY” might be modifying the entire phrase “PARK POLICE” if “PARK” means “public park”. Other than as an illustration of unclear writing, though, this example isn’t very funny.
Well, there was this cattle rancher. He decided to give control over the place to his sons Billy Bob, John-Boy and Melvin. He also renamed the ranch “Focus,” because it was where the sun’s rays meet.
One making the rounds in the mid 1980s concerned the corrupt Borough President of Queens, NY, Donald Manes, who ended up committing suicide.
What did Henry the Eighth, Rock Hudson, and Donald Manes have in common?
They all fucked Queens.
That may have been a Howard Stern joke.