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There is no legal maximum limit. However, each country requires that large amounts of currency (or currency equivalents) be declared to the proper authorities upon entry and exit, with harsh penalties for failure to make such a declaration.
For the United States, this amount is $10,000. Even though you will not pass through a customs check on your way out of the country, it is your legal obligation to go to the customs office and make a declaration if necessary. For the United Kingdom, the amount is β¬10,000. If you're flying through any other countries, you should review the relevant rules, as there may be requirements there too.
However, traveling with that much cash, or even large amounts significantly below the declaration limit, is inadvisable. It could easily be lost or stolen. It's also unusual behavior, and the authorities could suspect that you are involved in criminal activity, especially if you cannot readily prove a lawful source of the funds. Large amounts of cash could attract the attention of US law enforcement, who might seek to seize it. While this is all rather unlikely, it all makes the use of bank accounts a far better choice.