For any particular bank, it depends on the stability of its reserve during the process of credit creation. For the banking system, the expansion is 1/RRR. In Ch. XI of Mystery of Banking, Rothbard demonstrates that even with competitive banks, which are restricted in their issue of fiduciary media by redemption of their money substitutes by non-clients, the existence of a central bank will result in credit expansion from fiduciary issue as the central bank expands the reserves of the banks.
To use your example in Rothbard’s framework, suppose Customer X of Bank A deposits $100 cash in his checking account and the RRR=0.10, then the (minimum) the bank can expand its money substitutes by credit creation is $90. This assumes that Bank A makes a loan to a non-client, who will not accept a check drawn on Bank A but only cash. Then Bank A lends $90 in cash and is holding $10 against $100 deposit of Customer X. But the borrower either spends the cash or deposits it in his bank. If the cash is spend, the merchant then deposits it in his bank. Bank B, then, gets a cash reserve of $90 and credits Customer Y’s checking account with $90. Bank B can then lend $81 in cash and keep $9 reserve against the checking account of $90. And so on, until the entire banking system will have expanded checking accounts by $1,000 from the initial $100 increase in cash reserves.
Any particular bank can expand further than the minimum if it makes loans to clients instead of non-clients. Clients are those people willing to accept the bank’s checking accounts as a medium of exchange. So the maximum that Bank A can expand with $100 additional reserve and a RRR=0.10 is $1,000. Since, in this case, it does not need to lend any of its cash reserve in the process of crediting credit. The borrowers are willing to accept checking accounts drawn on Bank A instead.
Whatever might be the case for any particular bank, the banking system can expand by 1/RRR even in the case, as Rothbard demonstrated, where each bank can only expand by 1-RRR.