What minimum gauge wire can use in extending circuit for ceiling fan?

I had asked about a ceiling fan installation in my house, and I have a follow-up question. I have to run wiring across the ceiling beam from an existing fan. That will put them on the same switch. The easiest way seems to be running some thing wire from the existing fan connection, up through it's hole in the beam, across the beam, and down the fresh hole.
I can't use too thick of a wire for that connection, and I think very thin wires as run into the fan itself would work. However, I don't know if there's a major safety issue there. While it's just a fan, it will still run often.
I recall there's an issue of codes when extending connections. The alternative would be a heavy gauge wire up where the wire first comes out of the ceiling drywall. There isn't any slack there, and I see it being a real PITA to extend in that space. Furthermore, I believe codes state I'd have to make an easy access to that connection if I make it. This wouldn't be a big deal if I ran from the existing fan connection.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
The minimum size building wire allowable is 14 ga.

Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

HomeOwnersHub.com is a website for homeowners and building and maintenance pros. It is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.