I wish I had a way to take a picture because it's hard to explain. I've never seen a fig branch do this before. I was out this morning with my cup of hot coffee walking around my yard looking at stuff. I have an old espaliered fig on my front fence and was looking at it and noticed there is a branch about the diameter of a pencil, maybe a little thicker, and it had what I can best explain as two (much smaller) branches fused to the side of it. The branch tips weren't of equal length. There was the longest one, then the next one was a half inch or so further down the stem and then the last one was a bit further down than that. And they were fused by what I can only call 'webbing'. Like when someon has webbed fingers or toes, KWIM?  I think I'm going to snip that branch off and try to root it. It would make a neat looking trunk. 

Any ideas what could cause something like this? I'd post a picture here but am not sure how. Do I need to use a photo hosting site like Photobucket, or can I upload directly from my computer?

You can upload directly but it needs to be less than 1mb in size

Fusion of branches with each other is common in nature when two branches rub against each other wounding the bark eventually, over time, fusing with each other sometimes creating funny shapes.

Just what Ottawan mentioned, I see this on many street trees here.
If it was not so cold out in my yard and snow covered i could show example on my Elm tree.

It wasn't two separate branches that had fused together. It is one single branch.  Maybe I'll try to get a picture if I can figure out how. It's neat looking, regardless!

See single plant fusing:
http://en.wikipedia.org/wiki/Tree_shaping