Any way to update OSX 10.4's Bash version?

Do you have gcc and other tools installed? You can check with which gcc and then gcc --version. If you don't, you can install these from the Developers Tools DVD that came with the system, or you can download from Apple or gnu, but it might be tricky to get a version that will run.

Once you have gcc and the other command line tools installed, the next challenge is to obtain a bash source that will successfully compile and build on an old OS. I'm going to try this myself for 10.5 and 10.6.

Note that you don't need to worry about having a vulnerable Bash if the machine doesn't run a web server and isn't running services exposed to anyone but you. If you're the only user, and you don't run web sharing or other external services, you don't need to both.

As a stopgap, the easiest way to patch your system with a newer version of bash would be to use the version (4.3.30) available at the TenFourFox blog:

http://tenfourfox.blogspot.com/2014/09/bashing-bash-one-more-time-updated.html

The author provides step-by-step instructions for installing this version of bash.

Once you've gotten that preliminary patch successfully installed, it would be wise to compile and install the most up-to-date patched version of bash for your system. As others point out, you can either compile and install bash 2.05b (the version that came with Tiger) with all of the latest patches, or the very latest version with all the most recent patches (4.3.30). Either way, you'll likely need to compile and install these yourself.

If you do feel comfortable with compiling and building these yourself, then Tigerbrew will likely be of great help in accomplishing this. With Tigerbrew you can not only download and build a far more modern version of the gcc compiler (v4.9.1), but also bash 4.3.30. (Note that you'll need to have Xcode 2.5 installed before you install Tigerbrew.)