Are there some websites where the browser never prompts to remember your password? It might not be the browser's fault -- it could be the webpage telling the browser not to save the password. If you don't like having to remember these or write them down, you can use the Autocomplete = on Chrome extension to allow the browser to save the password.
Note: Can cause the browser to remember past input on other textfields as well; sometimes this can be distracting.
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