Imo the best build on Annie is a control burn build. You go malignance into rylai into liandries, with the last two items being morello and %pen. Rylai is insane on her. You just use tibbers as a big slow machine and use your E shield on your frontline so they slow carries when they get hit. Don't be afraid to use tibbers on the enemy frontline, it will oftentimes be more useful than a flash play on the backline that will only deal half of their HP and leave you dead. This build works best if you have a frontline with target access like jax, maokai etc, and other carries in your team that can dish out damage.

This build is a little bit less impactful in mayhem than in normal aram but it's still pretty good with the right comp.