I have a 13 and a half year old laptop, that has had a 100% dead battery for the past 10 years, and it has used on a consistant basis that entire time.
Most devices with rechargeable batteries have failsafes that prevent the battery from being overcharged, and in the case of dead batteries, not charged at all.
You are perfectly safe to continue to use it while plugged in all the time.
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