Most of the big banks at a certain fund level will reimburse ATM fees at will, and those can be held with the bank's investment arm, so you can put your stock there and get reimbursements. Wells Fargo has given me the highest ATM withdrawal limit, $5,020 which even many casino ATMs won't let you hit, although I am no longer at the highest level as they require $250k in assets for Premier. (Prime at $20k gets you one free withdrawal a month with reimbursement and combined with WF airport ATMs is all that I normally need.) Bank of America is $2,010 and their waiver level is at the Platinum level for $100k but the nice thing is once you get the status, you keep it for twelve months regardless. Chase is $75k although the Sapphire Checking offer is weak and their off site ATM limit is a pitiful $1,000 unless you call in advance to fix it. Citi is at $200k and they also do the $5,000 limit if you have the highest end account.
Schwab and Fidelity have unlimited ATM reimbursements, but their daily limits are $1,000 which are low unless you call, and then they only go to $2,000 or so. I personally dislike carrying cash and prefer to take it out of an ATM in Vegas for recordkeeping purposes, and at least it gives me a stopping point once the $2,000 is done so I can determine whether I want to continue or not.