Earning 50% and then losing 40% is equivalent to losing 10%. The naïve calculation you're arguing against would yield a gain of 10%. kgwgk didn't make that mistake.

The definition of expectation is what it is :-)

I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we lose everything but there is a zero probability of infinite gains that makes up for it!

1.05^n is > 1, with positive finite expectation at every finite time step.

Why do you say almost sure to lose everything?

For any threshold (say $1, assuming you start with $1mn) and any confidence level (say 99%) there is a number of flips n such that you have over 99% probability of holding less than $1 after playing n times. On the other hand, the expected value of your wealth at that time is 1.05^n million dollars.

This is well-stated.

In fact, an even stronger statement is possible: If you start with $1MM, eventually you will fall below $1 and never go above $1 again no matter how long you keep playing.

And all of that not just on a fraction of sequences, but with probability 1.

True, as I said in a previous comment we have “almost sure” convergence to zero (i.e. the sequence will converge to zero with probability one).