Clean build enabled: removing old files from D:\\BuildAgent\\work\\[...]
This causes a full checkout on every build, which takes time, and automatically rechecks the "clean all files before build" in the build configuration even if it was unchecked before. Note that the plugin isn't even used in any build. Its mere presence causes this behavior. Uninstalling the RedGate plugin it from the TeamCity server resolves the problem immediately.
From TeamCity developer (at http://youtrack.jetbrains.com/issue/TW-23391):
Please contact RedGate guys with this problem. Most likely somewhere in the code they have smtg like: BuildType.setOption(BuildTypeOptions.BT_CLEAN_BUILD, true). Not sure why they do this by they obviously forget to call BuildType.persist() so that this setting is changed on disk too.