LibreOffice is superior to the other freebie options in my books. It can both open AND save to .docx/.xlsx - OO cannot save to .docx or any MS formats. That's usually important when trading documents back and forth, and the primary reason why OO isn't too useful to me, should I decide to dump MS in the future. (currently running Office 2016 - trying to avoid subscriptions!)
There is also a LibreOffice for iPad. (and OO)
As for your original question re: installing MS Office 2016 (which is what you are now using) on a third machine, I would say probably you won't be able to install it on a third machine, but it really depends on what you got and how. "Downloaded" doesn't tell me much. If you purchased the perpetual licensed version of Office 2016, generally that is only good for ONE install (yes, MS has changed that - it used to be that MS Office could be installed on 3-5 machines depending on what version we're talking about). That said, I have a license for MS Office Professional Plus 2016 that we got via my husband's job and it installed on both the laptop and the desktop. However, that was under a very different licensing model and can't be assumed to apply to the home/student versions which all say very clearly "ONE install"... Normal retail Office 2016 and Office 2019 perpetual licensed versions only permit installation on ONE machine (making them very pricy at $170 per person in Canada) - you can get 3 years of Office 365 for that and it can be used on up to 5 machines in a household. Note: if you stop paying, you can view but not edit your documents.
If you downloaded your version of MS Office 2016 (which, btw, is not up to date) via some third party then all bets are off - a lot of them are much cheaper, but they're not legit and whether things can be updated, re-downloaded etc. is anyone's guess. I've seen reports of major fails when any attempt is made to update software obtained this way.