Science and medicine are probably my main interests, so I have a peculiar (and possibly :autism

fascination with this sort of stuff.
In terms of medical oddities, the first case that springs to my mind would probably be that of Robert Wadlow, an American man who, standing at 8 feet and 11 inches tall, was the tallest person ever known to have existed.
His tall stature was the byproduct of a malfunctioning pituitary gland, which caused him to grow not only a lot faster than his peers, but it also meant that he never stopped growing at all. Unfortunately, back in the 1930's when he lived, there were no forms of medical intervention which could have helped him, and so he died of complications associated with his condition at the age of just 22.
By all accounts he was a kind-hearted man, and apparently very strong up until the last few months of his life. He was already 7 feet tall by the age of 12, and by the time of his death he was wearing custom-made size 37 shoes.
I don't know if extreme longevity would be considered a medical oddity, but if it is, then there is no way I couldn't mention Jeanne Calment.
A French woman aged 122 years and 164 days old upon her death, she is officially confirmed as the oldest person ever to have lived, and the only person to have officially lived beyond the age of 120 (something which had previously been considered biologically impossible by some). Perhaps her greatest claim to fame outside of her longevity was the fact that she was the last living person to have met Vincent Van Gogh in person, whom she described as "very ugly", "ungracious", and "impolite" in an interview she gave about him 98 years after his death.
She was apparently fond of cigars, and smoked them every day throughout most of her life.
As for scientific oddities, I think the most breathtaking example I could cite would be UY Scuti, the largest star thus far discovered.
This red supergiant has a volume that is roughly 5
billion times greater than the volume of our Sun, and if it were at the center of our own solar system, it would stretch out as far as Jupiter. If you keep in mind that the Sun is roughly one million times the size of the Earth, you might gain some perspective as to how large this thing really is.