It is still an interesting example. Following your suggestion, which is a good one, the player's holdings at time N, starting with $1, are:
H = [1.5 or 0.6] * [1.5 or 0.6] * ... } for N factors
To make it into a more familiar process, take logs:
log H = log([1.5 or 0.6]) + log([1.5 or 0.6]) + ...
Now we have a familiar random walk with drift, and our intuition is much better-developed for this process. In particular, the mean
E log H = Sum{1..N} E log([1.5 or 0.6])
= Sum{1..N} [-0.023]
= -0.023 N
< 0
So the expectation of the log earnings is negative. In fact, because of the drift, sample paths of log H will be unbounded below with probability 1. This nontrivial fact (from the basic theory of random walk) will determine how the process sample paths (even outside the log domain) "look" - they will always eventually squash towards 0. Every single one.
But the expectation of the earnings themselves (as several comments have shown) is > 1. Apparently, without the compression given by the log, the upward excursions given by lucky bets outweigh the unlucky ones.
We can look at this discrepancy ("good game in original units, bad game in log units") the other way. Even though the sample paths eventually go to zero, in that case, you only lose $1. Whereas many other sample paths will make large excursions upward before coming to zero. So the expectation of the non-logged process adds up over many losses of $1, and a few low-probability gains of much more.