There’s a Henry VI irregardless.
Nitpick: The Man in the Iron Mask is the last few chapters of The Viscount of Bragelonne. V of B is a very long novel, and English-language publishers usually split it up into three or four sections. When split three ways, the sections are titled The Viscount of Bragelonne, Louise de la Valiere, and The Man in the Iron Mask. When split into four sections, one of the middle two is titled Ten Years Later.
My nominee is the House series:
House.
House II: The Second Story, which has no connection to the original story, except that it also takes place in a magical house where the kitchen door does not always lead to the kitchen.
The Horror Show, which has no connection to either of the first two, but was released in the UK and Australia under the title House III.
House IV, which is actually a sequel to the first movie.
Did the original Star Wars have “A New Hope” at the top of the scroll, or was that added later?
Here’s a weird example I found while googling around: Surf II. There was never a Surf I. Just to make things even more confusing, it was also marketed with a subheading: Surf II - The End of the Trilogy.
Which would have followed the same numbering pattern as Blackadder if they’d gone with Shrek Forever After’s original title: Shrek Goes Fourth.
No, it was not in the original version in 1977. “Episode IV: A New Hope” was added to the opening crawl in the 1981 rerelease of Star Wars.
Han: Whoosh!
Greedo: Bantha pudu!
28 Days
28 Days Later 
The 1979 film When a Stranger Calls had a made-for-TV sequel in 1993 called When a Stranger Calls Back. I assume that the franchise will be continued at some point with When a Stranger Leaves a Message and When a Stranger Sends You a Text.
Did I miss someone posting Hot Shots! Part Deux?
And, hot off the presses: I actually found one that satisfies the question in the OP. If recent news stories are to be believed, the sequel to 21 Jump Street, set for release in 2014, will be titled…
(drumroll…)
If they made a third (which I had hopes for) it was supposed to be called Shanghai Dawn.
Which almost exactly mirrors:
The Black Adder
Blackadder II
Blackadder the Third
Blackadder Goes Forth
If only that were even more true than it is.
Then the bafflingly inconsistent:
The Fast and the Furious: Tokyo Drift
Fast and Furious
Fast 5
Fast and Furious 6
Meanwhile, when the Tim Burton Batman series came about, there was some nerd talk that it ought to be:
Batman
Batman Returns (Re2rns?)
Batman Triumphant
Batman Forever
…but they went the boring route instead.
Batman Fi…re Joel Schumacher already.
Not mentioned yet.
The Whole Nine Yards
The Whole Ten Yards
and
Meet The Parents
Meet The Fockers
Little Fockers
Very funny ha ha. That’s like the Encyclopedia Britannica I once owned that had successive entries for Malcolm II, Malcolm III, Malcolm IV, and Malcolm X. I’ve always wondered what happened to Malcolms V - IX…
I disagree. The only real reason for those numbers was to indicate the moves were sequels.
Thanks to everyone for all the contributions, even though most (virtually all) did not answer the OP. But it was an entertaining discussion anyway, and that’s what we’re really here for.
KiDULTHOOD and AdULTHOOD
I’ll note that 22 Jump Street and The Whole Ten Yards are exactly the kind of thing I’m looking for. The 7*N UP series is close but not quite there. The same for Airport <year number>.
I was really hoping at the time that JPII would have gone with Pope Paul John Paul. The Fibonacci Popes!
I’m the first to mention Manhunter, Silence of the Lambs, Red Dragon, Hannibal and Hannibal Rising?
The Addams Family and The Addams Family Values
The Brady Bunch (1995) and A Very Brady Sequel (1997)