There are lots of articles and blogs out there explaining various ways to correct this problem. The reason I'm writing this small post is because I've seen and heard of many (recently) that have been trying to resolve this very problem in their SharePoint 2013 environment and have been through most of the common resolutions but one thing managed to elude many. The reason this problem has been coming to the forefront lately is because many are now migrating their SQL implementation to SQL Server 2014. SQL Server 2014 opens up many new features and amazing improvements which I won't even begin to touch on here but interaction between SharePoint 2013 and SQL Server 2014 is a little rocky out of the box. In order to get the two of them to play nicely you'll need to install the April 2014 Cumulative Update full package (be sure to get all 3 files or it will fail) during install. Remove your User Profile Application and the database associated with it and recreate UPA. OR....install the June CU which includes a fix for an issue the April CU creates related to authentication. I'll let you do the research on that.
Hopefully this will at least provide admins with the elusive fix for SharePoint 2013's inability to start up it's User Profile Synchronization service after installing or migrating to SQL Server 2014.
What's new in SQL Server 2014 [http://msdn.microsoft.com/en-us/library/bb500435(v=sql.120).aspx]
SQL Server 2014 Home Page [http://www.microsoft.com/en-us/server-cloud/products/sql-server/]