As of April 11, 2023 ChatGPT users on either free or plus plans can get an export of their ChatGPT prompt/response history and data by doing the following:
Sign in to ChatGPT at https://chat.openai.com
In the bottom left of the page click on Settings
Free plan
Plus plan
For "Data Controls" click "Show"
In the Data Controls menu which appears click on Export data
In the confirmation modal click Confirm export
You should get an email with your data
Note: The link in the email expires after 24 hours
Click Download data export to download a .zip file will be downloaded including your chat history in chat.html as well as your other data on file with us