I don't think it is possible without SSH. For what it's worth, I have GoDaddy deluxe shared hosting on Linux and I was able to install
Python as follows:
Set up a new directory in /home/"your username":
mkdir ~/python27
cd ~/python27
Download and unpack Python files:
wget https://www.python.org/ftp/python/2.7.12/Python-2.7.12.tgz
tar -xzvf Python-2.7.12.tgz
find ~/python27 -type d -exec chmod 755 {} \;
cd Python-2.7.12
Configure and install Python:
./configure --prefix=$HOME/python27
make
make install
Update your default Python path by modifying your bash_profile file:
vi ~/.bash_profile
Add these lines to bash_profile:
# Python 2.7
export PATH="$HOME/python27/bin:$PATH"
Confirm that the changes took place:
python -V #old version
source ~/.bashrc
python -V #new version
Install easy_setup:
wget https://bootstrap.pypa.io/ez_setup.py
python ez_setup.py
Quit SSH, log in again, and install pip:
easy_install pip
You can now add libraries as required. You can access the new version in your python applications with:
#! /home/"your username"/python27/Python-2.7.12/python
The following links were all helpful, but I needed to combine various steps for it to work for me:
http://geeksta.net/geeklog/python-shared-hosting/
https://www.godaddy.com/garage/tech/config/how-to-install-and-configure-python-on-a-hosted-server/
https://gist.github.com/BigglesZX/1610950