I'm surprised no one mentioned the Hawaiian Earring:
It's path-connected but not semi-locally simply connected (because any small neighborhood of the origin must contain a non-contractible loop). This implies many interesting properties, which make it a great counter-example. For instance...
- The Hawaiian Earring cannot have a universal cover.
- The Hawaiian Earring is not a CW-complex, although it is a compact, complete metric space
- An example of a space which is semi-locally simply connected and simply connected but is not locally simply connected is the cone on the Hawaiian Earring.
- For many years people thought the fundamental group was always a topological group. This turns out to be false, thanks to the Hawaiian Earring. There's a nice post about this here on MO
- This question is Community Wiki for a reason. I'm sure there are other examples of conjectures the Hawaiian Earring has disproven, so please add them!