I have a limited amount of free disk space left on my machine, (around 9 - 10 GB), and I would like to try out the latest Ubuntu 16.04 LTS on my computer, so my question is that, Is it possible to install Ubuntu 16.04 LTS with less than 25 GB of disk space (which is stated on the official documentation, as the recommended disk space for installation)?
25GB is the recommended available space, however the minimums are stated in section 3.4 of the installation guides. Visit https://help.ubuntu.com/16.04/installation-guide/ and select your machine architecture first. AMD64 with the desktop has a min of 10GB HDD.
A minimal server installation of xenial requires 400MB of disk space. The standard Ubuntu desktop installation requires 2GB.
So, technically, you should be able to install it on a far smaller drive (unless the installer refuses to do so, I didn’t try it), but of course you may easily run into out of disk space problems which easily play havoc on linuxes. So keep an eye on that!
You state in your question that you have about 9-10GB of disk space.
If you plan on running the Ubuntu Desktop, you must have at least 10GB of disk space. 25GB is recommended, but 10GB is the minimum.
Unless you can meet that 10GB minimum (and no, 9GB is not 10GB), you should not be using Ubuntu on that small a space, and should probably be cleaning off other stuff from your computer to make more space for your system. (Very few systems can operate with just 9GB or 10GB free, if the disk is large - the rule of thumb is once the drive dips under 10% of available free space, it's going to slow down considerably).
Source: https://help.ubuntu.com/16.04/installation-guide/, and knowledge gathered through trainings and experience