Long Live the Cookie!
I've got to say, I find the opposite [ie: cookies which expire after a short time, for no good reason] to be far more annoying. Thus requiring me to hunt around for my login details in the middle of interacting with a site I *thought* I was already logged into.
Amazon are probably the most annoying example: Their site will greet you by name, show you your recommendations and recent purchases and allow you to add items to your basket... right up until you hit the 'Checkout' button, at which time it's suddenly *"Sorry. Who are you? Log in please!"
eBay used to be similarly annoying but at least have had the decency to re-word the tickbox on their login screen to say "Keep me logged in for two weeks" now, which at least allows you to prepare yourself for being unceremoniously dumped on your arse on the doorstep, in a fortnight's time.
By far the most wall-headbuttingly annoying example ever though is Delicious.com, which arbitrarily logs you out at seemingly random intervals. Few things in life can equal the sheer joyous user experience of painstakingly filling in the details [description, tags,etc] of a website you want to bookmark, using their official bookmarklet, only for it to tell you [after submitting, FFS!] that you need to login. And, of course, after doing so you are returned to the bookmarklet form, which has now been wiped clear of all the info you just entered.
[That last one had me on the brink of insanity, til I eventually discovered Pinboard.in which has managed to allow me to stay loggged in continuously for several months so far, without it apparently 'breaking the internet']
—8000 year cookies? I'm all for them, in the right circumstances!