The DLC for Saints Row: The Third is pretty goofy. A nerd uses cloning technology to create a massive, monstrous replica of Johnny Gat. An asshole director hires you to make "Gangstas In Space," using unwitting extras to try and kill you for real "because it's cheaper." And the Genki Bowl gives you a few extra minigames to play as part of Steelport's inexplicably popular homicide-based reality game show.
The best part of the DLC was probably Sad Panda's Skyblazing. In it, you jumped out of a helicopter and had to fly through rings and intermittently land on rooftops to slaughter mascots with a chainsaw. It was completely different than anything else you'd previously had to do, but its combination of over-the-top violence and casual disregard for the physical laws of the universe let it fit right in.
The other notable thing about the DLC is that one of the "Trouble With Clones" missions gives you superpowers. I'm pretty sure that at the time it felt like a nice change of pace, but in retrospect, I couldn't think of it as anything but a preview for Saints Row IV. Which is good, because it whet my appetite for my next game, but also bad, because I can no longer appreciate it on its own merits. I guess that's just one of the consequence of getting older, though.
To sum up - Saints Row: The Third is the entry of the series I regard as the most essential. I don't want to call it my favorite, because the games each have such a unique feel to them that it's difficult to judge them against each other. However, it does serve as a bridge between SR2's "serious crime drama set in a ridiculous world" and SR4's "throw everything in the pot and hope for the best" approach.
I think, at this point, I have played Saints Row: The Third so many times that I'll never have to again, but it's such a fun and easy experience, that I'm sure I will anyway.