Actually, Third was supposed to introduce Prince Artie as a new running character, but when the sneak audience preview for #3 was a train-wreck disaster and most hated him, the studio quickly retooled their plans, took it back to the shop, and cut to the ending:
According to the studio's own pre-3 hype, #4 would have "father-figure" Shrek helping Artie in search of a bride (and the Little Mermaid parody), and coming home to get the big news about his own ogrelings in the climax. #5 would presumably have the New-Dad Calamities of an ogre and three babies, and #6 would be the big wrap-up we got.
(Which is why they went back after the preview and crammed the babies into Third at the last minute while they could, to save time on two films.)
Now this I would have probably more than happy to have sat through. But it seems that I am in the minority of liking the Prince Arthur character and plot. At least now I know why the plot was rushed through and why the damn pregnancy and baby storyline was jammed into the plot.