They will raise the transaction fees so high that your BTC isn't even enough to pay the transaction fee, so it will be impossible for you to spend them on chain.
What might that threshold be - 10 BTC, 100, 1000...?
I reckon base it on the percentage of total supply held and how that distributes within a typical power-law (Zipf's law) distribution.
I think you won't be able to hold on chain BTC unless you own at least 100 BTC and perhaps the threshold will be much higher. It depends how centralized is the control of the economy that is settled in Bitcoins. If it becomes centralized into 10 whales, they can make huge transactions with huge fees that block all other transactions.
At the point where there may be only a few million Bitcoin-holding entities, wouldn't anonymity be broken due to blockchain analysis methods combined with the reduced user base? Or would agreements be adhered to at the top, creating a glass ceiling barrier to the Bitcoin billionaire gentleman's club?
There are only ~2000 billionaires in the world. I think it will be a much smaller demographic than a few million, maybe 100,000.
They can mix their transactions with CoinJoin to obscure from the public (the jamming issue I pointed out to @gmaxwell won't be a problem for their club where everyone is known to each other). I don't think they will be allowed to obscure from each other. The elite want to know what is going on with their underlings.
Thus you can see how the all seeing eye takes form with Bitcoin and why I say it was created by the global elite and why I was curious what symbolic (even unwittingly) or actual role John Nash played in it.