Can I install Homebrew without sudo privileges?

Yes.

I modified the install script to not use sudo and to use a directory of your choice. https://gist.github.com/skyl/36563a5be809e54dc139

Download that, set YOUR_HOME in the script to the absolute path. chmod +x the script. Create the YOUR_HOME/usr/local directory. Then, execute the script.

./install.rb

In .bash_profile, I set (I'm not positive this is important, pretty sure):

export HOMEBREW_PREFIX=/The/path/to/YOUR_HOME/usr/local

Now, I can:

brew install wget

Make sure the bin directory, YOUR_HOME + /usr/local/bin is on your $PATH.

which wget

No.. Unless you do significant surgery.

The reason is that Homebrew strongly insists on installing packages into /usr/local. In fact, even if you forced it to install somewhere else, you are likely to break dependencies when you use brew install to install packages. Most if not all of these packages are pre-compiled and linked expecting to live in /usr/local.

The reason for this insistence is that /usr/local is precisely where POSIX recommends that stuff like this gets installed. In order to create /usr/local Homebrew needs temporary admin credentials to create the directory and assign ownership.

This, in turn, is what allows you to install anything else without elevating credentials.

In case somebody is still looking for this in 2020 and beyond, yes, this is possible without root privileges:

mkdir homebrew && curl -L https://github.com/Homebrew/brew/tarball/master | tar xz --strip 1 -C homebrew

The above allows you to install homebrew anywhere. Make sure that you add homebrew/bin to your $PATH.

More information about this alternative installation method (and source): https://docs.brew.sh/Installation#untar-anywhere