The latest and last official OS running on Note 3 is Android 5.1.1 and capable of running 6 and 7 smoothly via custom roms.
Android 8 have some errors on cam and network issues while 9 facing problems such as Camera , Gestures , UI lagging. Can push to 6 and 7 via custom rom without a problem but 8 and 9 can't be handled by the device. Android 10 unstable version via custom rom is available which is unofficial.
I prefer not using custom rom as it may effect device somehow. The toll impact would be great enough to clean whole phone!! Impact results switching to new phone. Better no more than Android 6 or 7 is recommended.
p.s. Kudos for your Note 3 . Still breathing eh! (kidding) #TrueSamsungnoteUser. thanks.
