Just browsing Amazon, there are a number of USB powered small, inexpensive grow lights. Pick one that fits your space and give it a try.
They are not battery powered, they have a cord and plug into a USB port. You could leave it plugged in and put the plant on your desk. They work fine.
I've never come across one powered by a rechargeable battery. I suppose you could try plugging it into a portable battery charger. I don't know if this could keep the light on all day, and you'd have to recharge the portable charger nightly. I'm not sure this would work, and it still would not be cordless.
What are you trying to grow?
