What material is best for building a time capsule which is supposed to last millions and millions of years?

Lots of excellent answers, so far. I'd use an outer casing of YSZ and an inner lining of gold, with the air inside replaced with one of the noble gasses. It almost doesn't matter which one, none of them are likely to react with the contents. However, contents can react with themselves. The interior will need to be cooled to the lowest temperature that will, itself, not cause damage. I see no way of keeping it that way for millions of years, but you might be able to reduce damage long enough. Which, really, is all that's needed. I'd therefore have the outer casing as the inner casing of what's ultimately going to be a large thermos flask. You want to coat it with something extremely reflective. Ultrapure aluminium or ultrapure silver should suffice. Reactivity doesn't matter because the gap will use a noble gas for innertness and it's going to be as close to hard vacuum as achievable anyway. The inner lining on the outer shell of the vacuum flask wants to be one of these new ultrablack materials. The next layer should be YSZ again. This should be in a web, again with an approximation to a vacuum with the only gas molecules being inert. I'm bored of noble gasses, so will use SF6 instead. Sulphur Hexafluoride is fun, provided it is on the INSIDE of an accelerator. You can use it anywhere I've wanted an inert gas. The web can be almost anything that doesn't react with itself. It's just a shock absorber. The web should not connect directly to the capsule, but rather to a sphere (probably YSZ) that contains the capsule. The capsule should be weighted and free to rotate, so that no matter how the ground shifts, the capsule retains a fixed orientation with respect to the centre of gravity and is self-righting as required. Outside that, a shell of YSZ inside a granite sarcophagus inside that silica-toughened concrete that occasionally gets mentioned as having phenomenal tensile strength, inside another granite sarcophagus. Actually you want this as a five layer sandwich, not three. To last a million years, you want it somewhere where tectonic activity has been and gone. In other words, on the side being built up. The continent's have lasted hundreds of millions of years, so you know that there are places that will survive just one million. The shock absorber will handle moderate quakes, preventing damage getting through to the thermos flask. Now, to make this slightly cleverer, I'll explain why the outer part is a five layer. The middle layer is going to be made up of randomly shaped pieces. A cross between a macroscopic quasi-crystal and art deco. The edges will interlock, but between the edges you want a compressible inert fluid. Doesn't matter what. The idea is simply to prevent the outer casing being completely destroyed in a freak event. It has to be able to shuffle shockwaves around, so that even if the inner and outer parts of the sandwich were mashed, the integrity is retained. Because nobody will remember where they put it, you should bury it with a black monolith with the sides having a 1:4:9 ratio that is permanently magnetized.

Just as important as the material of the time capsule (probably more so) is the location. Ideally you'd place the time capsule inside the rim of a crater on the Moon, Ceres, or some other geologically dead outer Solar System body to protect it from solar/interstellar radiation. Obviously retrievability then becomes an issue, but it's a trade-off you have to make. I mean erosion happens on the Moon, too. The existence of lunar regolith proves this (imagine how much worse things are on the Earth). On the Earth, my hunch is that the only place that actually lasts through geologic time without subduction back into the mantle is the core of one of the cratons. There are cratons that are almost as old as the Earth itself. You'd want to bury it deep inside. In either case (Earthbound or in outerspace), you might want to surround the time capsule with, say, a TON of highly enriched uranium (or anything else that doesn't occur naturally). It'll act as a long-lasting, super obvious homing beacon so that future civilizations could pinpoint it as an artificial structure pretty easily. They'll also be able to do some simple math to arrive at when exactly the time capsule was constructed.

You want a material that won't oxidize or is already oxidized.  Also, you want incredible abrasion resistance to defend from Mother Nature's favorite mode of wearing things down: slow and steady with sandy river water (think Grand Canyon).  Finally, I would want an opaque material because a couple million years of UV exposure will probably damage the contents.  I think I agree with Achilleas - go with YtZ.  Yttria-stabilized Zirconia is an incredibly tough, dense, hard white ceramic that besides being used in nuclear reactors is also used also in wet media mills. In these machines, thousands of YtZ beads are stirred with carbide agitators at thousands of RPMs inside jar of abrasive water. The purpose is to grind down the particles in the water, but without wearing out the beads.  This material performs very, very well due to its outstanding properties mentioned.

On the order of millions of years -- there are processes which make such a guarantee not possible.  These include: * Metasomatism, or changes to the chemistry of the material.  This is often due to the action of groundwater, and certainly influenced by the chemical species nearby. * Petrogenesis - the dirt nearby all will turn to rock. * Metamorphism from burial. * Tectonic activity.  This can turn anything to bits. I would favor making a capsule out of glass -- since that is pretty well inert.  Metals will interact easily with groundwater.  Plastic will degrade from sunlight or natural radiation present from the breakdown of hornblende inclusions in nearby rock, for example.  But glass is chemically stable in the presence of water, and won't break down so easily from radiative exposure.  Moreover, any chemical changes will probably result in exchanges in the glass itself -- and these may not impair the integrity of the material at all.  I'd make the glass thick, and look into various glass compositions. I'm not sure how such a capsule should be sealed ... but probably best if it can screw together. PS It's not a bad idea -- This reminds me to make a time capsule to fill with photocopies of science articles and class notes I'm no longer using -- to save them for posterity! PPS I just remembered another flaw with metal -- specifically that microbial action sometimes works to break down the material.  As an example, for heavy metals, current remediation practice involves placing microbes in the contaminated area.  These microbes generally are not designed in the lab, but rather taken from another site with such a contamination -- one wherein researchers have noted the levels of the contaminant dropping.

Plastic.  Bugs don't like it. Water bounces right off of it.  It's hard as a rock. Polycarbonate plastic has the advantage of being transparent. Being organic, it will eventually degrade, especially if exposed to sunlight.  So, if you've got an unlimited budget, you go with gold.  Gold doesn't petrify; it doesn't rust; it doesn't decay. Gold is pretty damn immutable. Gold is also pretty soft, so you either need to leave it some place where it won't be punctured, or you wrap plastic (in an airless environment, but well structured) in a good thick sheet of gold (to keep out the air and light) and the two will survive pretty much anything short of a direct attack. A good compromise choice would be some kind of tempered glass, which is chemically highly resistant and (if you get the right kind) insanely strong.  And a lot cheaper than gold.

A million years is very, very doubtful because of unknowns but here goes... Titanium.  Resists high temperatures, UV, acids, seawater and many other chemical reactions.  It doesn't rust and will flex under stress without cracking.  Aircraft grade titanium beats the best steels on just about all points.  So your capsule is safe unless it gets into a volcano eruption or nuclear war... All organics, including plastics, degrade under UV light after just a few years. Diamond is abrasion resistant but is is brittle and also burns. Glass almost does the job except for its tendency to crack under stress. More exotic metals like tungsten may do the job but at a much higher cost and marginal improvement.

I was going to say Gold-Plated Titanium, but since this was more or less taken. I'll go with a space craft designed to orbit the sun for 1M years and then land. I'm sure you can design a careful orbit (may be polar?) that will allow the space craft to orbit nearly indefinitely without hitting anything.

Millions of years is tough. You have to take in the possibility of volcanic activity, tectonic shifts,and even a small amount of erosion like 1mm/1,000 years is equal to a meter of wear every million years. So fused quartz maybe? Wear resistant, crazy high heat resistance. Or go the old fashioned way and use Granite, just use lots of it. It is cheap and abundant, so future generations would not likely  salvage it for material. And you could make it massive. Face it, after millions of years everything will erode somewhat, so carve your time capsule out of a 100 foot block of granite so you have plenty of extra to erode away.

Given that money is no object the best material I can come up with is based on its extreme chemical stability is Tungsten.  It will last and indeterminable amount of time with extremely low reactivity and virtually no vapor pressure so there would be no sublimation deterioration. But the bigger question is what are you going to put in this capsule that you expect to last that long?  Most things of interest to future civilizations would be materials we commonly use and they are general chemical reactive and would oxidize or deteriorate.  The capsule would have to be put some place where there is a vacuum and where the temperature approaches 0 degree Kelvin for the more delicate materials inside survive extreme lengths of time.  We are also assuming that someone that far in the future would ever find it.  And even assume there will be someone / something around that would even care to know!  After that period of time the significance is likely to be totally lost.