Titanium
We need
@Yellow Saddle to explain.
Titanium is a much-hyped material. Probably the most hyped-up material other than carbon fibre.
It is steeped in mystique, right from being named after the children of Uranus and Gaea in Greek mythology, through to scarcity during the cold war. It is said that Ronald Reagan and Michael Gorbachev sealed the end of the cold war deal on a golf course somewhere by teeing off with a set of drivers made from titanium - supposedly repurposed missile shell skin. I dunno, but that's how deep the titanium lore goes.
As an engineering material, it has its place, but it is not the be-all of materials. As for any material, it has its place and titanium's place is not in bicycle frames or sprockets. Perhaps it has a role to play in the odd nut and bolt but for the rest, aluminium and steel is better.
Material strength isn't a single entity or the final the deciding factor for whether or not something should be made from it. For instance, I write with plastic ball-point pens. Plastic is strong enough. You can make a stronger pen from titanium or carbon steel, but what is the point?
Titanium is strong, but steel is stronger. Further, it isn't all that hard either. Steel is harder. On the Rockwell scale, carbon steel has a hardness of 71. Titanium (depending on the alloy) is about half that. But, hardness has no function in bicycles other than for chains and bearings. Hardness is useless in frames, even counter-productive.
For its weight (all being equal, Ti is about half the weight of steel and double that of aluminium), it is pretty strong. However, when used in bicycle frames it isn't all that effective because the tubing cannot be drawn thin enough to make the frame rigid. Aluminum can be drawn very thin and produce great, stiff frames. Just look at an oversized aluminium (such as Cannondale) frame or admire the gossamer thinness of a Heinneken can. Titanium can't do that and that's why Ti frames are relatively thick and small diameter, making them more flexible than steel or alu of equal weight.
Further, it has some serious weaknesses. At weld seams, it's brittle toughness is low enough to lead to frequent cracks in frames. Steel performs far better at welds and aluminium, for its weight, even better, since heat treatment effectively fixes any weld-induced weakness.
One thing Ti has going for it is that it doesn't rust. Whoopee. Another thing is that it welds beautifully, producing fantastic smooth, flat welds. These are much nicer than the equivalent job in aluminium. To show off the welds, the frame is often not painted, lending it even more high-tech mystique.
It is difficult to machine because it readily work-hardens. As the tool cuts into the billet, it hardens in the region just ahead of the cut zone. This blunts the tool but also generates heat, causes smearing, heat-induced weakness and gives the machinist a general hard time. But note, per se, it isn't harder than steel. Don't confuse Ti-nitrate with Titanium. The former is that golden stuff they coat machine tools with. That is hard.
As for all the other magical properties assigned to titanium bicycles - it is mostly nonsense. It doesn't have some sort of magical vibration damping properties. If it feels "plush" it because it flexes like a noodle or, some sort of auditory feedback fools you into thinking it is plus. It isn't. It transmits as much vibration as does any hard material, be it carbon, alu, steel, salt crystals or beer bottles.
Some high-end sprockets in bigger sizes are made from Ti - Dura Ace, Campag Record, etc. They simply don't last and I doubt anyone actually replaces them with more Ti sprockets once the cassette has worn out. They also cost the earth.
Someone mentioned a chain gouging a Ti chainstay. That's because a chain is much, much harder than titanium suitable for frame material. It will eat through it like a rat trapped in a cheese-flavoured cardboard box. This leads back to the question of sprocket longevity.
I am surprised that watch casings for heavy watches like Rolex and other oyster-style casings aren't more freely available Titanium. I assume is it because stainless steel polishes up nicer and resists scratches better. It is the perfect material for that being corrosion proof and light. A chunky watch in lightweight material is nice to wear. Sektor used to make anodised aluminium watches in the 1990s and those looked the part without the weight of steel watches. Titanium has found its way into some watch casings but I never see them in action.
On a bicycle I'll tolerate titanium on the seatpost, to prevent it from freeze-corroding with aluminium or carbon, I'll accept it on some low-stress bolts such as cable anchors and brake pad retainers, headset bolts and lever clamps. It isn't suitable for crank bolts, solitary stem bolts or frames. There are better materials for those applications.