Is there a Free-To-Play item that my character can hold in Runescape that increases experience output, even slightly?

You can have urns in your inventory that will grant bonus experience in fishing, cooking, mining, smithing, woodcutting and prayer. The lower level urns are free to play items.

Also, cooking on a bonfire will slightly increase the cooking xp gained, as will burning logs on a bonfire instead of burning them normally. The lower level bonfires are free to play.