The memory limit is already quite big and I could see it from the log file. If it's not your host stopping the backup process, it's something in your computer / browser such as an antivirus or Internet security application. If you are not running such a thing or if the error persists even though you have deactivated your antivirus / Internet security application there are two more troubleshooting steps:
1. Make sure that your computer does not go to sleep mode or disconnect from the Internet and you do not close the browser tab or window where the backup is running. Javascript running on your browser triggers each backup step, checks the server response, updates the interface and repeats this loop until the backup is complete. If your computer goes to sleep, you get disconnected from the Internet etc then the chain of backup steps is broken, the browser reports an error and you get a log file similar to the one you sent us.
2. If this doesn't happen your host is lying. Most likely the first level support technician answering your support request isn't privy to the exact configuration of the server or is not allowed to disclose the measures taken by the host to protect against denial of service attacks. In most cases you can escalate the issue and talk to a real system administrator who will be able to explain why the server kills the backup.
FYI, there is no shared hosting on this planet not enforcing resource limits that is still in business. If they enforce no limits apart from PHP memory limits someone could buy the cheapest shared hosting plan and have the CPU and hard disk on the server work at full capacity, effectively bringing down the server. Having someone from your host telling you that they don't apply any limits is either a blatant lie or they an admission that they are absolutely suicidal and about to go out of business. That's why I'm sounding so sure. Plus the nearly eight years of experience supporting Akeeba Backup (called JoomlaPack back in '06) on shared servers which made me aware of all that stuff to begin with :)
Nicholas K. Dionysopoulos
Lead Developer and Director
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