I'm using an Exchange 2013 SP1 environment with almost no customization. Only 2 servers exists - one holding CAS+MBX, and a second one being an MBX. No DAGs, balancers, etc. Mapi over HTTP is not enabled. The default self-signed certificates are used (no new certificate was installed, nor any self-signed certificate manually installed on any server/client). A mailbox is provisioned on a database located on the first server. Outlook is configured for the corresponding user on a client machine and started. Everything works just fine, with the 'Outlook Connection status' window showing 2 Exchange Directory + 2 Exchange Mail connections. Authentication is NTLM. Ports for all 4 connections are 6001 - which hint that Outlook Anywhere is indeed used.
From time to time, the familiar "Security Alert" comes up warning about the self-signed certificate, but this is usually traced in my experience to the various services Outlook is using, that are running on HTTPS (OAB, EWS, Availability...).
Here we find that in Exchange 2013 "Outlook Anywhere is enabled by default, because all Outlook connectivity takes place via Outlook Anywhere". Then here it's stated that "Outlook Anywhere won't work with a self-signed certificate on the Client Access server". I remember the latter being true against Exchange 2010 instances, but seems not to be the case in Exchange 2013 anymore. Unless I'm missing something, from the standpoint of a default installation, the 2 articles contradict each other.
Second issue - even though Outlook is set for "Negotiate" in its Security setting, it looks like the Kerberos preferred option is never chosen. Would it have to do with the self-signed certificate and Outlook Anywhere ?