Run a script only at the very first boot

No. But you might want to place your script in /etc/init.d/script, and self-delete it:

#!/bin/bash

echo "Bump! I'm your first-boot script."

# Delete me
rm $0

Combining the first two answers Assuming you name your script /usr/local/bin/firstboot.sh put it at the end of /etc/rc.local (this scripts runs on every boot) the scripts goes like this

#!/bin/bash

FLAG="/var/log/firstboot.log"
if [ ! -f $FLAG ]; then
   #Put here your initialization sentences
   echo "This is the first boot"

   #the next line creates an empty file so it won't run the next boot
   touch $FLAG
else
   echo "Do nothing"
fi

Create a tracking file when the script runs. If the file already exists, exit the script.

In my case, it was building a custom system and I had to refuse to use cloud-init and placing a script under /etc/init.d/script didn't work - so I used systemd.

FILE /etc/systemd/system/firstboot.service

[Unit]
Description=One time boot script
[Service]
Type=simple
ExecStart=/firstboot.sh
[Install]
WantedBy=multi-user.target 

and

FILE /firstboot.sh

#!/bin/bash 
SOME COMMANDS YOU WANT TO EXECUTE
systemctl disable firstboot.service 
rm -rf /etc/systemd/system/firstboot.service
rm -f /firstboot.sh

Then enable

(sudo) chmod +x /firstboot.sh
(sudo) systemctl enable firstboot.service

This one works just fine for me.

I'm surprised at the results I'm seeing for searching for a well-defined and supported Ubuntu "first boot" hook. Seems like the Red Hat / Fedora / CentOS crowd has had this nailed for over a decade. The closest Ubuntu equivalent seems to be oem-config-firstboot.

The idea of simply performing an rm $0 will work. But, technically there are some interesting semantics involved. Unlike most other script interpreters under Unix a shell script is read and processed one line/statement at a time. If you unlink (rm) the file out from under it then that the instance of the shell that's processing that script is now working with an anonymous file (any file that's open but unlinked).

Consider a file like this:

#!/bin/bash
rm $0
echo "I've removed myself: $0"
ls -l $0
cat <<COMMENTARY
   This is a test.
   I'm still here, because the "here" doc is being fed to 'cat'
   via the anonymous file through the open file descriptor.
   But I cannot be re-exec'd
COMMENTARY
exec $0

If you save that to something like rmself.sh and (hard) link that to something like tst then running ./tst should show something like this as output:

$ ./tst 
I've removed myself: ./tst
ls: ./tst: No such file or directory
   This is a test.
   I'm still here, because the "here" doc is being fed to 'cat'
   via the anonymous file through the open file descriptor.
   But I cannot be re-exec'd
./tst: line 11: /home/jimd/bin/tst: No such file or directory
./tst: line 11: exec: /home/jimd/bin/tst: cannot execute: No such file or directory

Now there are some odd possible corner cases with regards to symlinks and cases where the script was invoked as a bare name (forcing the shell to search the $PATH for the script.

But it seems that bash (at least in version 3.2) prepends $0 with the path if it searched the path and otherwise leaves $0 set to whatever relative or absolute path was used to invoke the script. It doesn't seem to do any normalization or resolution relative paths nor symlinks.

Probably the cleanest "firstboot" for Ubuntu would be to create a small package (.deb) containing a script to be placed in /etc/init.d/firstboot and a post-install script that uses update-rc.d to link that into runlevel 1 (/etc/rc1.d) (using a command like: update-rc.d firstboot defaults) ... and then have the last line perform a deactivation or delete using something like: update-rc.d firstboot disable

Here's a link to the Debian update-rc.d HOWTO