In your login shell's profile files you can set up some things that you will use during your session, and which only need to be done once. Some ideas:
create a temporary file containing the IP address you connected from, later on you can include it in some scripts setting firewall rules.
run ssh-agent, ask for your SSH keys, and store the SSH agent environment variables in a file.
if that is a limited machine, and your co-workers want to be aware of each others logins, write(1) messages informing them of your login.
In a non-login shell's files (.bashrc) you should configure your shell:
Fancy prompt
set aliases
set history options
define custom shell functions
export environment variables, (maybe PAGER, EDITOR if system-wide settings suck)
load ssh-agent variables saved in .bash_profile
Usually, you would include .bashrc from .bash_profile with the following. Then login shell gets all your customizations, and .bash_profile file does not have to duplicate things that are already in .bashrc.
Best Answer
The default shell in BusyBox is ash.