Dumbledore's chocolate frog card states that he discovered the twelve uses of dragon's blood. That sounds like a pretty foundational set of discoveries. How one goes about figuring out the magical properties of a substance, I don't know, but at least in this case it must have been fairly difficult. After all, brilliant alchemist Flamel didn't figure them out during his incredibly long life. No one did.
Dumbledore was only born in the 1840s or so, and probably could not have done any research of that caliber until he was at least, say, twenty (since he was supposed to have been an exceptionally brilliant wizard). Discovering magical properties is research into essential natures of substances. Potions and Transfiguration appear to alter something's essential nature, and Dumbledore was the Transfiguration professor, so it seems Dumbledore's forte is discovering and manipulating the essential natures of things (as opposed to charming them to sprout legs or change colors, or to successfully handling Venomous Tentaculas). Between that and his brilliance, it's no wonder he ended up partners with a noted alchemist.
But about that partnership: the card also states that Dumbledore and Nicholas Flamel are research partners. Researching what, pray tell? Isn't the Philosopher's Stone supposed to be the pinnacle of alchemical research? Flamel already figured that out centuries ago. So what research was he doing after that? If he was intelligent and talented enough (and spiritually pure enough, if JKR is following tradition - but she doesn't always) to create a Philosopher's Stone, why, in all those centuries, did he not figure out at least a few of the uses of dragon's blood? Dumbledore discovered the twelve uses, not just twelve of many, if I'm reading that right.
Is Dumbledore really so brilliant that he outclassed in some ways a man five centuries his senior who had managed to create the only known Philosopher's Stone in existence? Or had Flamel settled down by that point and just turned into the "idea guy," letting his younger research partner do the actual work?
Or is discovering the properties of dragon's blood so fundamentally different a process than creating the Philosopher's Stone that Flamel just wasn't up to it? It's hard to imagine, but maybe. Perhaps Dumbledore and Flamel had complementary skill sets. I'm having trouble seeing a huge gulf there, though. (There is the fascinating, but probably far-out, idea that maybe whatever Dumbledore was doing was something a spiritually pure person couldn't do... but that purity doesn't seem to be required just to drink the Elixer once available, so it isn't really necessary.)
So what kind of partnership was this? And did they do any interesting research between 1958 (or so - whenever the card was printed) and 1991? Considering that's the period when Dumbledore was trying to figure out what Tom Riddle had done to himself in his quest for immortality, maybe they were reverse-engineering the effects to find the causes. (In a purely theoretical way?) Though that's a pretty short list, right? How many paths to deathlessness are there?
I'm probably overthinking this or overlooking something obvious, and it might come clearer in the morning. But it does appear to be a very curious partnership.
Actually, there is one thing I can think of which may be an example of the Dumbledore/Flamel partnership: the enhanced blood protection charm Dumbledore placed on baby Harry which keeps Harry safe from Voldemort so long as he resides with his mother's kin. The protection is built on two things: blood (possibly Dumbledore's specialty, seeing as he figured out dragon's blood when no one else could) and a mother's loving sacrifice (more spirit-related, which is possibly Flamel's speciality, seeing as he created the Philosopher's Stone). That charm could be the perfect combination of their talents.
We've only seen or heard of witches and wizards using time-turners to go into the past, haven't we? Hermione skipped back an hour or two at a time to take extra classes, and Harry and Hermione went back three hours to save Buckbeak and Sirius. Not being seen is essential, to keep one's past self from doing something rash and/or to keep from introducing a time paradox. (Not that Harry playing James helped that issue much.)
Can time-turners also send people into the future? I ask this not just because it's a gaping hole in our understanding of time-turners, or because I can imagine a Back to the Future version of HBP where Snape says, "Nobody calls me chicken," but because it would provide one more way to keep the pattern of Dumbledore Explains It All intact. Can you imagine a Harry Potter book without a rundown from Dumbledore at the end? We could always get it from his portrait, or from someone else standing in for Dumbledore, but it's just not the same.
Just imagine: it's time for the wrap-up, and right on cue, Dumbledore appears. But that's impossible! Harry protests. "Professor, you're dead!" "Ah," says Dumbledore, "I rather thought I would be. At the moment, however, I have sent you to fetch your Invisibility Cloak - which of course you already have with you - before we embark on our journey to Lord Voldemort's sea cave."
There are all sorts of problems with this scenario, of course. For starters, maybe time-turners simply can't send you into the future. And how would Dumbledore know when to show up? Unless he has been using his time-turner a lot. (Which opens up all sorts of possibilities for chronological confusion throughout the series, and gives a reason for why his hair turned white between 1945 and 1957 other than stress or genetics...)
As I said, it's a crack theory. (I've got another, even crazier one, but I'm still piecing it together.) It is a good amusement for a rainy day.
ETA: Checked, and Hermione does say that wizards have killed their past or future selves by mistake. Does she mean that the time-traveling wizard goes into the future and kills himself, or that the time-traveling wizard is the future self who is killed?