Suppose that with each game you have probability .7 of winning and .3 of losing. Say you bet half your money in each game. You start with $Y_0$ = 1. Let $Y_n$ be your bankroll after $n$ games and $$ Y_{n+1} = Y_n \cdot W_{n+1}.$$ with $$ W_i = \begin{cases} 1.5 & \text{with probability}~.7 \\ .5 & \text{with probability}~.3 \end{cases}.$$
It follows, $$ Y_n^{1/n} = \left(\prod_{i=1}^n W_i \right)^{1/n} \implies \log Y_n^{1/n} = \frac{1}{n} \sum_{i=1}^n \log W_i = \mathbb{E} \left( \log (W_i) \right) .$$ Now, $$ \mathbb{E} \left(\log W_i \right) = (.7) \log(1.5) + (.3)\log(.5) $$ and hence \begin{align*} ( Y_n)^{1/n} \to e^{ (.7) \log(1.5) + (.3)\log(.5)} &= e^{ (.7) \log(1.5)} \cdot e^{ (.3)\log(.5)} \\ &= (1.5)^{.7} \cdot (.5)^{.3} \\ &= 1.0788. \end{align*}
This is not a valid application of the strong law of large numbers... what's the sample mean here? Also what is $W_i^n$ ... there are $n$ different possible values of $i$ here... you can only take them to be all the same after you take the expectation value.
The trick here is to take logs. You have $Y_n = \prod_{i=1}^n W_i$ so $$ \log Y_n^{1/n} = \frac{1}{n} \sum_{i=1}^n \log(W_i).$$ Now, this is a sample average of $\log(W),$ so is something the law of large numbers applies to since $\log(W_i)$ are i.i.d. and integrable. The mean of $\log(W)$ is $$ \frac{1}{2}\log(1.5) + \frac{1}{2}\log(0.5) = \frac{1}{2}\log(3/4),$$ so $$ (Y_n)^{1/n}\to e^{\frac{1}{2}\log(3/4)} = \sqrt{3/4}. $$
Note that this means, somewhat counterintuitively, that $Y_n\to 0$ almost surely, even though $E(Y_n)\to 1.$ In fact, by inreasing the win probability slightly, you can find an example where $E(Y_n)\to \infty$ but $Y_n\to 0$ almost surely.
Edit whoops, I didn't see that the probabilities of going up vs down were $.7$ and $.3,$ not 50-50. I will leave it up to you to plug those in. Since this is purportedly the Kelly criterion, I'm guessing $Y_n$ will converge to $\infty,$ not zero, almost surely, i.e. the answer will be $>1.$
Edit 2 Actually, the Kelly criterion for these odds is betting $40\%,$ but betting $50\%$ isn't too greedy and still gives positive growth.