A lot of personal information about ownership / beneficiaries needs to be declared to the banks and tax authorities.The way to bypass this is to open a company, you own the shares of the company, and the company invests in the stock market.
This applies to all jurisdictions.. and gets tougher depending on how secretive the country where the company has been formed is. For example if you form a company in a jurisdiction which allows for bearer shares - most banks will refuse to deal with you, unless like pin mentioned you have a few billion dollars (or atleast 100s of millions) and can find a dodgy / greedy banker to help you.
BVI etc which are more "regulated" are also problematic in terms of opening bank accounts at top tier banks.
The sooner you accept that some people are more equal than others the better your life will be.I believe that this is how billionaires do it.
Like others have said ... gold in a safe deposit box is probably your best option.