Another thing they're quite keen on is power supplies, and their ability to start a fire (or not). They'll want to demonstrate that the amount of power available from any power outlet is inherently limited, and although there are various techniques to achieve this, a (UL recognised) PTC fuse in series with any power supply can be a straightforward way to achieve compliance.
It's worth having a talk with a UL representative. When a product is granted a UL mark, it brings with it a level of insurance cover; UL are declaring that they believe your product to be safe, and they agree to underwrite the cost of a successful claim against you in the event that your product causes harm. Being an insurance company, they'll want to ensure that the product you're actually making is identical to the one they tested, so they carry out regular factory inspections and may ask for other documentary proof that the parts being used are the same.
It's quite an overhead, and can be a very costly process - but that's one reason why it's nice to be doing the development work, which is a one-off process, rather than being responsible for the ongoing compliance overheads once a product is in production.