Does Windows 10 have a built-in SSH server?

Short answer: Not yet, but it's coming.

Long answer: It's not actually part of Windows 10, but rather a new feature Microsoft is adding to PowerShell. Microsoft is working with and beginning to contribute to the OpenSSH project. This will effectively put an SSH server on Windows but you'll have to turn it on and have the latest version of PowerShell. I have not heard any release date for PowerShell with the OpenSSH functionality mentioned yet.

Additional reading arstechnica.com

Official Microsoft Announcement msdn.com

Update: The newest technical preview versions of Windows 10 Anniversary update have two services "SSH Server Broker" and "SSH Server Proxy". These are both built-in to windows 10 Build 14366 and above. It is still unclear about the client however and I have been looking unsuccessfully for Microsoft documentation about these two services...