While playing Coup I have often pondered whether you can target yourself with the assassin. This might seem counterproductive at first, but actually has several very exciting use cases:
Turn Skip. Allow time to challenge, and block with a Contessa. At a very high price of (most certainly) giving away some information on your cards and running a high risk of challenge due to board information known by others and paying 3 coins, you could potentially earn a turn skip.
Bluff opportunity. Especially when paired with a Contessa, but also when maintaining the expectation of using the Contessa afterwards, and in tight end-game situations, using the assassin against yourself can provoke others to challenge.
Threat management via loss. As a subclass of aforementioned strategies, you may want to give away information, or lose either a card and/or coins to be perceived as less of a threat.
Chaos. I love Coup so much just because it lets you to play with the minds of others. What could throw someone off balance more than this?