"Otherwise what I was mostly trying to suggest was that the banks anticipate the fact that the central bank won't let them double the supply of money and factor this into their loan and deposit pricing. The idea is that the current amount of deposits is not so much based on the curren[t] supply of base but the supply expected in the future." (That was from commenter HJC, with my emphasis added and one assumed typo fixed).
Now that is what I call a real and important critique of the money multiplier, as exposited in the textbooks. Because the textbooks are implicitly assuming a permanent increase in the money base, but none of them AFAIK make that assumption really explicit and talk about the difference between the current monetary base and the expected future monetary base. And it is a critique that has important real world implications, like for the US right now. And it is us Market Monetarists who should be making that critique.
Compare two different cases:
1. The Bank of Canada announces it will permanently double the monetary base relative to what it would otherwise have done. It recognises that doing so will permanently (approximately) double the price level relative to the price level path that it would otherwise have chosen under 2% inflation targeting, and it wants this permanent doubling of the price level to happen.
2. The Bank of Canada announces that a computer glitch will cause the monetary base to double, but only for one month. Because the techies are absolutely certain that they can fix it, but are currently all on holiday. The Bank of Canada assures everyone that normal programming of 2% inflation targeting will resume shortly, and that it will take steps to ensure that this computer glitch will have zero permanent consequences for the price level, if necessary by tightening monetary policy in future to restore the previously planned path for the price level.
In the first case I would expect an (approximate) doubling of currency in public hands and doubling of deposits. The money supply would double, as would the price level and all nominal variables like NGDP. Nominal interest rates would rise temporarily, then revert to normal.
In the second case I would expect approximately nothing to happen. The banks would roll their eyes and sit on (approximately) all the extra base as reserves for one month. (Since Canadian banks normally hold very small amounts of reserves, because none are legally required, and the base is currency+reserves, the stock of reserves would much more than double.) The overnight rate would fall 0.25% to equal the deposit rate (the rate of interest the Bank of Canada pays on reserves), and one month interest rates on liquid assets would fall a little too, but approximately nothing would happen to loans and deposits and the stock of currency in public hands.
I think that the current US case is much closer to case 2 than to case 1. Yes, the US banks have been sitting on the extra reserves for much longer than one month, but it is the conditionality that matters more than the duration. As someone (sorry I forget who) once said: the Fed has put out a large punchbowl of free booze, but no individual bank wants to drink much unless the other banks drink too, and the Fed says it will take away the punchbowl as soon as they start drinking. The full punchbowl just sits there. The Fed would need to announce that it wanted the price level (or NGDP) path to be permanently higher to give them permission to start drinking. And if it did that, a much smaller punchbowl would work much better than the current uselessly large one.
Other critiques of the money multiplier totally miss the point. Like "loans create deposits!" or "base is endogenous!" or "banks don't lend reserves!" or "other things affect the money supply too!". This critique matters because the textbook exposition is designed to show (among other things) how the central bank's control over its own balance sheet (which is all it really controls) allows it to control the money supply, in the same sense that I control my car. (Yes, the position of the steering wheel is endogenous, given the bends in the road I have chosen to take.) The current state of the balance sheet matters less than the expected future balance sheet, and the expectations about the conditions under which the central bank will change that balance sheet.