Sorry for reopening this topic, but I must add a word of caution. Assuming the theme you're using isn't the default theme provided with the WordPress installation, upgrading WordPress, whether automatically or manually, might render parts of the theme unusable if the theme isn't compatible with the new WordPress version (which is true for any CMS). Same goes for the plugins. For instance, if a certain template tag or function in the theme (or plugin) is deprecated in the new WP version and replaced with another, then the part of the theme using this function or plugin will not work. Therefor, if I were you, I'd strongly discourage my clients from upgrading by themselves and even disable this option from being displayed in the admin panel. New WP installations do try to take care of compatibility issues, but I would trust that just to a certain point - I wouldn't expect, say, WordPress 5 to treat incompatibilities issues down to WordPress 3.
Note that you don't need to upgrade WP for the blog or website to continue functioning properly, so even if there is an upgrade available every month you don't need to upgrade every month. In fact, the website will continue working the same way it did so far even if you never upgrade. Of course it's recommended to upgrade when there's a security update, but other than that you can upgrade only when there's a major release or when minor updates have accumulated to a certain degree. So you might need to upgrade once or twice a year. Then you should check beforehand if each of the plugins works with the new version (often there's an upgrade for the plugin you should download separately), and if there are any changes you should make to the theme so it would work properly with the new WP version. You should also make a copy of the old WP version before upgrading, just in case you missed something and need to revert to the old version. And this is true for any CMS - upgrades should not be done by the clients unless they know exactly what they're doing.
Otherwise, if you let your clients upgrade by themselves, then at some point or another - and this is bound to happen, whether next month or two years from now - you will get hysterical phone calls right in the middle of the first night of that perfect romantic holiday you've been planning for and saving for for months, desperately crying about the website falling apart. And then, after canceling your vacation without a refund, getting screamed at by the missis or hubby and finally dumped, and after spending hours to find what could possibly go wrong and pulling half your hair out, while your clients is losing visitors by the minute, you'll find out it's only a minor change in one function that requires changing two lines in one of the plugins. A mess that could have been avoided if upgrading was done by someone who knows what he or she is doing.