TopHostingCenter is my host. Here is my conversation with them.
It looks like some issue with the wordpress. If you wish to upgrade the wordpress, kindly follow the below mentioned process.
1. Backup your database. Read Backing Up Your Database for a detailed explanation.
2. Backup ALL your WordPress files in your WordPress directory. Don't forget your .htaccess file.
3. Verify the backups you created are there and usable. This is essential.
4. Deactivate ALL your Plugins.
5. Ensure first four steps are completed. Do not attempt the upgrade unless you have completed the first four steps.
6. Download and extract the WordPress package from http://wordpress.org/download/.
7. Delete the old WordPress files on your site, but DO NOT DELETE
* wp-config.php file;
* wp-content folder; Special Exception: the wp-content/cache and the wp-content/plugins/widgets folders should be deleted.
* wp-images folder;
* wp-includes/languages/ folder--if you are using a language file do not delete that folder;
* .htaccess file--if you have added custom rules to your .htaccess, do not delete it;
* robots.txt file--if your blog lives in the root of your site (ie. the blog is the site) and you have created such a file, do not delete it.
8. Upload the new files from your computer's hard drive to the appropriate WordPress folder on your site.
9. Run the WordPress upgrade program and follow the instructions on the screen.
10. Update Permalinks and .htaccess. Update your Permalink Structure and merge the custom rules, if necessary, into your .htaccess file.
11. Install updated Plugins and Themes. Please review the list of Plugins that work in Version 3.0.5. Check for Theme Compatibility with 3.0.5 and ask your Theme author for any new version.
12. Reactivate Plugins
13. Add security key definitions to the wp-config.php file
14. Review what has changed in WordPress.
Please let us know if you need any further assistance.
To which I replied...
I see, thank you for the complete, if somewhat cryptic instructions, I will attempt to implement them if wordpress support itself fails.
Can you provide me with proof (I need not be able to understand it) that the problem with WordPress 3.0.4 being unable to auto update is definitively not a webhost issue? On my end no error is given, the process simply stops, do you have any more information as to where and how exactly it fails?
I will be opening a support thread on their forum in an effort to permanently correct this problem.
I find it hard to believe it's actually a wordpress problem (I could of course be wrong, I'm no expert), and so will they I suspect, so this proof is important as it will enable me to refute them when they just send me back to you.
To which they replied...
We checked the logs and we did not find anything in our server logs which makes it clear that the issue is not from the server. We will wait for the update from wordpress team and will proceed accordingly.
So how can I determine who's problem this is?