What a nihilistic challenge...
Alright: No Apollo-1 fire. Apollo has a successful *Apollo-7 flight, but then when the equivalent of *Apollo-8 rolls around, the command module is destroyed on the launchpad thanks to a problem due to the fact that the new safety features introduced in the wake of Apollo-1 weren't. While NASA struggles to respond, the Soviets try their own moonshot after a successful UR-700 test flight. This gets to the moon, but an airlock failure (a common problem on early Soyuzes) kills all the cosmonauts, after the USSR had already started broadcasting live footage.
The upshot of all this is that a moonshot is regarded as too dangerous/impossible, manned flight by extension, and everyone reverts to just using space as a place to put satellites with unmanned launchers.
Depressing enough for you?