despite the fact that...

[sunsunmoon]

▪ The law has yet to be passed, despite the fact that most people are in favor of it.
(From http://www.learnersdictionary.com/search/despite)

Can the fact that be left out?

[Rover_KE]

Yes, but the sentence would have to end '. . .despite most people being in favour of it.'

Rover

[sunsunmoon]

She wanted to let her fans know that despite she has been busy; she will be posting more video more often thanks to her upcoming album.
(From Rebecca Black Confirms Album Is Coming Out Soon In New Video)

Is despite she has been busy incorrect, but is despite her having been busy correct?

[Rover_KE]

Is despite she has been busy incorrect, but is despite her having been busy correct?

They are both incorrect.

'. . . despite having been busy' (which should be followed by a comma - not a semicolon).

Rover

[Bennevis]

She wanted to let her fans know that despite she has been busy; she will be posting more video more often thanks to her upcoming album.
(From Rebecca Black Confirms Album Is Coming Out Soon In New Video)

Is despite she has been busy incorrect, but is despite her having been busy correct?

"despite her having been busy" is correct.

[sunsunmoon]

Al Sharpton preached about the “dangers of inflammatory rhetoric” in The Washington Post, despite his being found guilty of defaming a prosecutor and helping incite a deadly race riot.
(From Government Fines Businessman for Creating Jobs)


Asher said the team was totally surprised by the result, despite his being a member of the winning team last year.
(From Perfect score for quiz winners | Otago Daily Times Online News : Otago, South Island, New Zealand & International News)

If despite his being and despite his being are grammatically correct, wouldn't despite her having been busy be correct?

[Tdol]

I think it is, though it sounds better without her. In the two examples above, it works better in the second where it helps clear up that he was the only one on the winning team the year before. In the Al Sharpton example, it adds nothing but wordiness IMO and would be better without it, though I wouldn't go as far as calling it wrong.