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Is there a way to export the entire browser history from Chrome under Android, in any file format (csv, xml, whatever)?
I have Chrome 42.0.2311.111 under Android 4.2.2.
This may be more than a year late to this question, but...
You can use Google Takeout to download a json file of your Android Chrome history.
You can also download your bookmarks, along with a lot of other Google data.
So what if it's more than a year late. It helped me, thank you for posting. I gave it a +1.
Chrome doesn't allow backup using ADB and there are no extensions available for mobile version so you should have root access before you proceed further.
Grab the History file from the location /data/data/com.android.chrome/app_chrome/Default/ into PC. A Root Explorer may help.
Install an SQLite manager such as SQLiteman or DB Browser for SQLite in PC.
Both has the option under menu bar to export the database into an SQL file or a particular table into CSV file.
Well at least you now know where the history is stored, so you may find a relevant software to directly export the database into a CSV or XML file.