We already know what can last for literally millions of years: Life.

Maybe we could encode some information in birdsong, tree rings, or the matting patterns drawn by fish in the seafloor. :)

https://www.youtube.com/watch?v=hpdlQae5wP8

This reminds me of Whisper, one of my favorite exotic alien concepts. The alien civilization in question has selected a planet with as stable environment as possible, and planted it full of genetically engineered grass. The sound of wind blowing through the grass creates an acoustic computer which hosts a virtual space its creators have uploaded themselves into.

https://www.orionsarm.com/eg-article/470007b39d192

I have never heard anything like that (using sound) before. Amazing!

A SF short story about a biologically encoded message from ancient aliens (written by one of the Autodesk founders): We'll Return, After This Message by John Walker

https://www.fourmilab.ch/documents/sftriple/gpic.

It's fascinating that in 1983 it was thought that a 768-bit RSA number would take 40 million years to factor. (It actually took 26 - RSA-768 was factored in 2009.)

The link seems broken? Should be: https://www.fourmilab.ch/documents/sftriple/gpic.html

Or directly encode the message in DNA itself. I suppose the difficulty would be devising an encoding with enough redundancy or other features to preserve the information content over many generations of evolutionary pressures.

Our plastic is going to be around far longer than us. Left by itself plastic decays on geological timelines.[1]

1. http://worldwithoutus.com/excerpt.html

If by "stay" you mean spread microplatisc, that carries no information, then maybe. The jury is still out, but it looks like it'll last a long time.

But plastic is not a good candidate to hold messages for a long time. It degrades in a few years into some unusable stuff. Glass is a much better candidate.

Unless a microbe finds a way to use it as fuel

There's a microbe for everything: https://cen.acs.org/environment/sustainability/Plastics-recy...

If you knew that a past civilization or even non-human intelligent species had been wiped out by war or climate change, would it change your actions?

I don't know there's much we could say that future civilizations would care to listen to.

It would certainly be big news.

Deep Time: How Humanity Communicates Across Millennia Paperback by Gregory Benford explores some of those questions. It comes into practical questions when trying to mark nuclear waste dumps.

Build large stone piles.

https://en.wikipedia.org/wiki/Great_Pyramid_of_Giza

The problem with that is, judging by our own behavior, an unknown ancient structure makes humans investigate it, try to dig it up, etc. Placing a marker isn't the problem, the hard part is communicating "nothing interesting, only death here, do not enter" in a way that someone in 10.000 years will heed the warning.

The solution is to build them everywhere and put the bad stuff in one of them. If you build enough then the chances of them excavating the one with nuclear waste is pretty low.