How to add an icon to the bash prompt

Actually, Yes, you can.

In recent versions of Bash, at least 4 (i could do it in 4.2 and 4.3), you can render emoji with the hex.

Use the echo -e flag.

paste an emoji you looked up in and do a hexdump to see what it's made of:

plasmarob ~ $ echo -n ""| hexdump

0000000 f0 9f 87 ba f0 9f 87 b8                        
0000008

And then take that top line and escape each hex pair with \x :

plasmarob ~ $ echo -e 'See? \xf0\x9f\x87\xba\xf0\x9f\x87\xb8'
See? 

I actually modified mine to be:

plasmarob ~ ⚡

So yes, come up with one like this and try adding it to your .bashrc or .bash_profile.

Edit: Something with SO or browser rendering may have changed because the flag in this post now renders as a "US" character. YMMV but i assume it will still work in the stated versions of bash.

Nowadays, you can add emoji if you have an emoji-aware font. I guess this wasn't a easily viable option when the question was originally posted

I wrote this blog post about it a couple of years ago.

I don't know about American flags, but export PS1="\360\237\232\251 > " gets a flag in your prompt.

I also wrote a shell tool to make printing the escapes for echo or shell prompt a little easier. It's called emo

Sorry, no. Terminals don't do graphics.

For a full description of what you can do, see the PROMPTING section of the bash(1) man page:

PROMPTING

When executing interactively, bash displays the primary prompt PS1 when it is ready to read a command, and the secondary prompt PS2 when it needs more input to complete a command. Bash allows these prompt strings to be customized by inserting a number of backslash-escaped special characters that are decoded as follows:

\a     an ASCII bell character (07)
\d     the date in "Weekday Month Date" format (e.g., "Tue May 26")
\D{format}
       the  format  is  passed to strftime(3) and the result is inserted into the
       prompt string; an empty format results in a locale-specific time
       representation. The braces are required
\e     an ASCII escape character (033)
\h     the hostname up to the first ‘.’
\H     the hostname
\j     the number of jobs currently managed by the shell
\l     the basename of the shell’s terminal device name
\n     newline
\r     carriage return
\s     the name of the shell, the basename of $0 (the portion following the final
       slash)
\t     the current time in 24-hour HH:MM:SS format
\T     the current time in 12-hour HH:MM:SS format
\@     the current time in 12-hour am/pm format
\A     the current time in 24-hour HH:MM format
\u     the username of the current user
\v     the version of bash (e.g., 2.00)
\V     the release of bash, version + patch level (e.g., 2.00.0)
\w     the current working directory, with $HOME abbreviated with a tilde (uses the
       value of the PROMPT_DIRTRIM variable)
\W     the basename of the current working directory, with $HOME abbreviated with a
       tilde
\!     the history number of this command
\#     the command number of this command
\$     if the effective UID is 0, a #, otherwise a $
\nnn   the character corresponding to the octal number nnn
\\     a backslash
\[     begin a sequence of non-printing characters, which could be used to embed a
       terminal control sequence into the prompt
\]     end a sequence of non-printing characters

The command number and the history number are usually different: the history number of a command is its position in the history list, which may include commands restored from the history file (see HISTORY below), while the command number is the position in the sequence of commands executed during the current shell session. After the string is decoded, it is expanded via parameter expansion, command substitution, arithmetic expansion, and quote removal, subject to the value of the promptvars shell option (see the description of the shopt command under SHELL BUILTIN COMMANDS below).

The \e, \[ and \] escape sequences deserve special attention. With these you can insert ANSI escape codes to command the terminal to change foreground color, background color, move the cursor, erase parts of the screen, and do other fancy tricks.

That is, for instance, how your prompt changes color. \[\e[0;31m\] sets the foreground color to red, and \[\e[0;0m\] resets it back to the default.