How high can they fly?

No particular species, just birds in general.
How high can a bird fly?

Pretty darn high.

I was going to mention the bar headed goose in the link by Arrendajo. The Planet Earth series has an episode with them migrating over the Himalayas.

A bird can fly as high as a woodchuck could chuck wood.(If a bird could do that)
(Your bird’s mileage will oscillate)

According to this, the highest recorded flight was a Ruppells Griffon Vulture and it was observed at 27,000 ft(!!)…so, pretty damned high.

Your link actually says 36,000 ft.