Arkham Horror
Can you assign damage or horror beyond an asset's health or sanity to it, and where does the excess go?
No. You can assign only as much damage or horror as the asset has room for, and the rest is suffered by you or goes to another asset you may assign it to. An asset is discarded once its damage or horror equals or exceeds its limit; exceeding it can only happen through other effects, such as one that lowers its health.