How to execute a script periodically without using crontab?

The modern option is to use a systemd timer unit. This requires creating a systemd unit which defines the job you want to periodically run, and a systemd.timer unit defining the schedule for the job.

Assuming you want to run the job as regular user, put these files in $HOME/.config/systemd/user:

my-job.service

[Unit]
Description=Job that needs periodic execution

[Service]
ExecStart=/path/to/your/script

my-job.timer

[Unit]
Description=Timer that periodically triggers my-job.service

[Timer]
OnCalendar=minutely

Then enable the newly created units, and start the timer:

$ systemctl --user enable my-job.service my-job.timer
$ systemctl --user start my-job.timer

To verify that the timer is set:

$ systemctl --user list-timers
NEXT                         LEFT     LAST                         PASSED UNIT         ACTIVATES
Wed 2016-11-02 14:07:00 EAT  19s left Wed 2016-11-02 14:06:37 EAT  3s ago my-job.timer my-job.service

journalctl -xe should show log entries of the job being run.

Refer to man systemd.timer for the many options for configuring timer behaviour (including randomised starting, waking the computer, persistence across downtime, timer accuracy, etc.), and to man systemd.unit for excellent documentation on systemd and systemd units in general.

You could run a script with at and have it reschedule itself as its final instruction. For instance like this:

$ cat > $HOME/my.job <<END
do this
do that
cat "$HOME/my.job" | at now + 5 minutes
END
$ sh ~/my.job

To stop the job, remove it from the at queue:

$ atq
5       Wed Nov  2 11:25:00 2016 a zwets
$ atrm 5

Or simply remove the file my.job so the next execution of cat "$HOME/my.job" will error so at will not be invoked and the rescheduling doesn't happen.