> Upon login, bash does not automatically execute my `~/.bashrc` script. Every other Linux distro I've run does this automatically, so I'm at loss to why Raspbian doesn't.

The first line of the default `.bashrc` should give you the answer:

    # ~/.bashrc: executed by bash(1) for non-login shells.

It's for non-login shells, you are expecting it to be loaded for login shell. And it's the same for any bash instance, regardless of the "Linux distro".

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> Is there another file that bash runs on startup?

Per `man bash`:

> When  bash  is  invoked  as  an  interactive login shell, or as a non-interactive shell with the `--login` option, it first reads and executes commands from the file `/etc/profile`,  if  that  file exists.   After  reading that file, it looks for `~/.bash_profile`, `~/.bash_login`, and `~/.profile`, in that order, and reads and executes commands from the first one that exists and is  readable.

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> what's the syntax?

Either of the files mentioned above is a bash script file. No special syntax is required.