How can I install MongoDB 3.X on Windows without admin rights?

Yes, it is possible to install the latest MSI (including the one with SSL) without admin rights via command line.

msiexec /a mongodb-win32-x64-3.2.5.msi /qb TARGETDIR="C:\MongoDB"

This will copy the binaries into C:\MongoDB\MongoDB\Server\3.2\bin

I dislike long paths like that, so I create a symlink inside the folder:

cd C:\MongoDB
mklink /j bin C:\MongoDB\MongoDB\Server\3.2\bin

That will create a soft link as C:\MongoDB\bin (which you can add to your PATH environment variable).

mongo --version
mongod --version

Both should return version 3.2.5.

You can do this with most packages, we have to do similar with Python 2.7 and Node 4.4.3 MSI packages on work computers that do not have admin rights.

You can download the "legacy" version which is the unsigned non msi version as a zip. The disclaimer is listed as

The 64-bit legacy build does not include SSL encryption and lacks newer features of Windows that enhance performance. Use this build for Windows Server 2003, 2008, or Windows Vista

The 3.0.5 version is https://fastdl.mongodb.org/win32/mongodb-win32-x86_64-3.0.5.zip