I can not run a php script on my current shared hosting accounts because of their php execution time limits. The script will run by crontab. I don't need to host a website. I don't need to run DNS, email and database processes. So, I may not need any control panel if you will help me to upload the script and set up the cron. 10 MB space and 1 GB bandwidth will be enough. I will buy a package less than $10 monthly.
Agreed but I wouldn't try that though. There's a good chance doing such is against the ToS/AUP of the host. Have you asked the host to change the script time limit? That's really the best bet as it gets them involved.
Well, from what you described it seems like you only need a vps to play with your script. Yes, like prilep said, you can find unmanaged vps for less then $10.