There is no really cheap bank when it comes to offshore banking. You must also take into consideration that it has become very difficult to open an offshore account anywhere. You best choice if you are going for cheap may be EMI accounts like Mister Tango, LeuPay etc.
Alternative real banks may be Cyprus but you will find it difficult to get your account opened there if at all.
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Disclaimer: Nothing I say should be taken as tax, legal or financial advice. Anything I say is for general informational purposes only. Always seek independent professional advice.