* It's very difficult to sell a market to zero, but impossible to buy it to infinity.

Has any stock or financial instrument ever been valued at infinity? I don't know enough about finance to rule that out and it seems unlikely but at least possible.

For instance here is a toy example of unbounded value: Consider a market for a stock in which the only market participants are two bots with the behavior that they trade the stock back and forth at an asymptotically increasing price. The buys are backed by loans from a zero interest government bank. Since the bank is efficiently hedged to losses. Both sides of the trade owe the bank, the seller can also pay the bank the money the bank needs to back the buyers loan. Thus, the bank could allow both these loans to go to infinity causing the value of the asset to approach infinity.

Similar things have happened with flash loans in the cryptocurrency space.

At what point does a finite increase of the price turn it from a finite number into infinity?

No. Money is a scarce resource and modern algorithmic trading has limits to prevent another LTCM.

What is the price of a share that isn't listed for sale?

Emptying out all the asks is a thing that happens.