Great question!
1. Bruce and Stephanie
Stephanie Brown is my favorite character, and her training, editorially mandated axeing or no, is still some of my favorite stuff. Her run-in with Zsasz at Batman's side is full of rich nuance and cheer - similar to Dick, but with her own unique flavor. She also has some of Jason's rage, which Bruce helps to temper. I personally reject the idea that she is the most underqualified - like n52 Tim, she was an accomplished gymnast before her time as Spoiler, and her skills impressed Selina Kyle in Catwoman #34, even after she was fired as Robin.
2. Dick and Damian
I dislike Grant Morrison's actual writing of these two, but the conception and execution of this duo is perfection. Whether it's Bryan Q. Miller in Batgirl, or Kyle Higgins and Scott Snyder in Gates of Gotham, the contrast between a lighter Batman and a darker Robin is genius.
3. Bruce and Tim
Some have said that Tim is too much like Batman to be as interesting as his predecessors, but I actually think that Tim, in both post-Crisis and post-Flashpoint incarnations, is actually very much different enough. Tim is a great detective, and works incredibly hard at the physical side of Robin, but I love his combination of loyalty and distance. Whether it's the concern and attention he shows to Bruce during Batman Eternal, or his determination to find the truth first in Bruce Wayne: Murderer, he's always struggling to find the right balance between his love and loyalty to Batman and what he stands for, and his own judgment.
4. Bruce and Damian
I totally agree that these two are too similar to really provide much contrast. I appreciate Bruce and Damian's growing together in Tomasi and Gleason's strong run, but they are both so selfish and similar that they don't provide the real counter to each other that I think a Robin really needs to provide for Batman.
5. Bruce and Dick
Confession: I've never really read anything of these two in continuity outside of Batman and Robin Eternal's flashbacks, not exactly the most stellar of examples. I love both individually, but I prefer Dick as his own man.
6. Bruce and Jason
Pretty much the same situation for me as with Dick as Robin, and similarly, I prefer Jason as Red Hood (except when written by Morrison. Because really?)
7. Bruce and Carrie Kelly
Great fun, but Miller didn't give Carrie enough focus for me to care. I do like their relationship in DKIII, but until that story's done, I don't know that it makes quite enough of a difference in rating.
8. Jean Paul Valley and Tim
Haha, this one's a joke, right? Did they ever actually work together as Batman and Robin? I liked their friendship both under Batman, but they were never really partners, I thought.
Bonus question; Which Dynamic Duo team do you think would make for an interesting pair that has never happened?
Cass and Steph, with Cass taking on the lead Bat role in Gotham, and Steph filling a strong partner role as Robin or Batgirl. I don't ship them romantically, but their friendship was consistently a fantastic thing (when Cass wasn't being terrible and knocking Steph out "for her own good.") The Damion Scott Solo issue where Cass trained Steph as Robin is really sweet, and their training sessions in Batgirl were adorable.
Steph and Damian - technically, it did happen! In Steph's Batgirl run, she acted as big sister in an incredibly adoreable (and effective) issue, and even got Damian to act like the kid he needed to be. This dynamic is sorely missed, and other writers have tried (with varying degrees of success) to fulfill it, such as Tom King with Duke Thomas, or Pat Gleason with Maya Ducard. None of them quite have the fury-tempered hope of Steph to contrast with the cruelly-accelerated cynical nature of Damian, though.
Every single Robin! In the Batman: The Brave and the Bold issue #13 where the Phantom Stranger brings all six Robins together to fight for Batman!