Thanks for filing the issue!

Checked the issue on reported chrome version 66.0.3359.117 using Ubuntu 14.04. We were able to see minimise button without any issues. Attaching the screen shot for reference. 
As the issue seems to be specific to Debian OS, hence forwarding it to Inhouse team for further triaging. Adding label"TE-NeedsTriageFromHYD".

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Also try: "Settings" > "Use system title and borders" > Disable

This problem may only be specific to Debian or may be specific to xfce or compiz compositor.

I am experiencing the same issue on Debian. I am running KDE 5 with KWin (so that probably rules out xfce/compiz issues). This problem started happening in the last 1-2 days, maybe after a package update.

I don't use the user feature, and the minimize button is gone even in Incognito where the user switcher does not appear

Chrome says: Version 66.0.3359.117 (Developer Build) built on Debian 9.4, running on Debian 9.4 (64-bit)

Packages: chromium 66.0.3359.117-1~deb9u1, plasma-desktop 4:5.8.6-1

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Unable to reproduce this issue on Debian Rodete with XFCE with chrome #66.0.3359.117 & #66.0.3359.139 as per steps mentioned in the commnents #0,#4

Attaching the screen-cast for reference. 

sky5564@ Could you please look into it and let us know your observations.

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The same problem on Debian9.4 KDE.  

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Chromium is reading your GTK settings for the frame button layout.  If you run an app with client side decorations like gtk3-widget-factory, you should see the same behavior.  To change this, you'll need to change your GTK settings.