The closest that you'll find is probably Rockruff, which can have Keen Eye, Vital Spirit, Steadfast (HA), and Own Tempo (event-exclusive). However, Own Tempo is technically a separate form from regular Rockuff, which only has the first 3 abilities. This means that you can't use Ability Capsules / Patches to switch between Own Tempo and non-Own Tempo the same way you could if it were an actual 4th ability on the same form.
If you want an even more niche case, the Litwick line used to have Shadow Tag as their HA, but it was changed to Infiltrator before the HA was made available. Since Litwick also has 2 standard abilities, you could argue that the Litwick line had 4 abilities, though only ever 3 at a time.
So no, there is no singular form of a Pokémon that has more than 3 possible abilities at the same time.
