Why is the Firefox tabs close button disappearing? How do I get them back?
I haven't looked at the Firefox source, but it's likely that the tab-rendering code removes the close buttons first if the tabs become too small. That is likely why they disappear when you open the 8th or 18th tab (as in your examples).
The response from Critical Failure is spot on, the rendering code removes the close button when the number of tabs is too high to save space on the screen.
There is an add-on for Firefox that allows you to control this behaviour: https://addons.mozilla.org/en-US/firefox/addon/8005
It allows you to have a close button on all tabs, only the active tab or neither.
This method also stopped working at some point, possibly with 29 (Australis) or 57 (Photon) UI. I am not aware of any alternative at the moment.
You can navigate to about:config and set the value browser.tabs.tabClipWidth to 1. You will need to restart the browser.
As others have suggested, the close tab button is removed if the tab becomes too small (narrow). This setting determines the minimum width of the tab in pixels below which the close tab button is not shown (140 is default). Setting it to 1 essentially means always show the close tab button, no matter how narrow the tab is. According to the linked article, 0 is not a valid number (but the button is 16 px anyway).
This setting has an effect in all versions of Firefox from 1.6a1 onwards (confirmed working in 13.0.1).
As of Firefox 23.0, this method no longer seems to work. They may fix it in a future update to the extension, so I will leave it here for now. The config option described above is still available.
An alternative (and the one that I am using) is the Tab Mix Plus extension. This extension includes many other possibly (un)desired features, and obviously isn't as clean as the setting above, but it is what I've been using (for other reasons, the close button was just a side effect).
In TMP's options, Display > Tab > Close tab button > on all.
Click for full size
Wilson's answer links to an extension that is no longer available. Both the config and TMP setting described in this answer achieve the same effect.