It makes sense that CAS servers would steadily fill the log folder in inetpub. What's not so clear is why the mailbox servers log folder is filling up as much as the CAS servers. Any ideas?

Because everything that goes through the CAS also goes through the mailbox server.

You can turn down IIS logging or deploy a script that cleans the logs up if you want.

Thx Ed.

I thought only the CAS servers handle all the client facing connections (email client, active sync, webmail) that generate this kind of IIS chatter.

We have another environment that is similar to this DAG but it's 2010 Exchange instead of 2013. On those servers the IIS log folder size on the MB servers are less than 1% the size of the IIS folders on the CAS servers. This is in line with what I expect. Why is the Exchange 2013 mailbox servers so different? 

Because the Exchange 2013 architecture is different.  In Exchange 2010, the CAS does all the web translation and talks to the mailbox store directly via MAPI.  In Exchange 2013 there is no MAPI and the CAS is just a protocol router.

Got it. Thx Ed!

You're welcome. Happy to have helped!