Oh, I have a strong contender.... and a sad story as well.
As many know, BBS's were big years ago, before the Internet took off.... Well, after getting a 14.4 modem for my miggy, I found at least 5 full-time Amiga supporting BBS's in Milwaukee alone. One of them was called PSA.... (Politics and Social Awareness, IIRC) it was run by then-Milwaukee County Supervisor Dorothy K. Dean.
In maybe 2006 or so, while chatting with her on the BBS, she said that the BBS (and the A2000 it ran on) had been on 24/7 for the last 8 years !! (Not sure if she had a UPS, I imagine so) I know her late husband set up an AREXX script, in WBStartup, to automatically run the BBS program if power ever failed. Sadly, when her husband (and tech. support) passed away, PSA went offline for good...
Personally, I only had my A1200 on constantly for about 12 days.... for a Lightwave .ANIM .... Psst, anyone have a spare Raptor laying around...
