Best way to periodically execute a PHP script?

You can do without cron using the following solution (PHP only): How to run PHP cron tasks without cron available

Not sure if this is the correct approach, but I used to just trigger a script when the first user visits the site, and then send a <meta http-equiv="refresh"... to the user for his browser to refresh the page. The original PHP script would still run on the server, but the user will not see it anymore.

Basically, something like:

if( check if the user is the first visitor today ) {
    set_time_limit(0);
    echo "<meta http-equiv='refresh' content='1;url=..." />"; // put your site baseurl in here

    ... run your scripts here
}

Just an idea. Might not work. Just try it out.

You can use an online cron service to essentially pretend like you have cron access.

Create php file with contents you would like executed

Free Cron Online Website

Set up your free online cron to execute that file every x minutes.