I think MaryPat meant Euro €.
I belong to 2 credit unions.
One has a $400 limit that has not changed in the 15 years I've been a member and absolutely will not up it for any reason. I've complained multiple times about how small that amount is, no matter what the exchange rate is. They don't care.
The other has limit so large that I've never reached it. You can guess which one I use to get cash.
I haven't experienced limits at cash machines associated with banks in Italy, but if those limits are per transaction but not per day it's possible that you could withdraw the cash in multiple transactions, from different banks IF your cash card allows that. See the link at the bottom for more advice on using cash machines on Europe.
What you definitely want to do is get the cash in local currency. Do not accept the DCC (Dynamic Currency Conversion) invitation. Here's why from the RS Travel Tip on Money:
Some ATM machines also offer DCC, often in purposefully confusing or misleading terms. If an ATM offers to "lock in" or "guarantee" your conversion rate, choose "proceed without conversion." Other prompts might state, "You can be charged in dollars: Press YES for dollars, NO for euros." Always choose the local currency.
And here's the link to the section on using cash machines: https://www.ricksteves.com/travel-tips/money/cash-machine-atm-tips