You purchased stock for \$1m. What is the probability that it is worth more than $30m after 10 years? The change in value of the investment each year is modeled as follows:
Divided by 2: 1/4
Remain unchanged: 3/8
Doubles: 1/4
Quadruples: 1/8
Where I'm at: I'm aware that this needs to be formulated into a normal distribution, but I'm not sure how to come up with the right formula, or how to correctly approximate it to normal. Would really appreciate some insights!
 A: No normal (even continuous) distribution would fit your evolution rules, which are clearly discrete (if only because $P(X_{t+1} = X_t) = 0 \not= 3/8$.
With your evolution probabilities, the best you can do IMO is to enumerate all possible values of the stock in year $n$ for $n \le 10$ and then compute the probabilities each year using a probability tree (this is Google's first result).
PS: you may want to enumerate values for $\log_2 (X_t/X_0)$ (if $X_t$ is your value in year $t$) so transitions from $t$ to $t+1$ are $-1$, $+0$, $+1$ and $+2$.
PPS: if really you wanted to fit a normal distribution, I'd suggest you compute the expectation and variance of your distribution for $X_t/X_{t-1}$ and use these as $\mu$ and $\sigma^2$ parameters for your normal dist. (with common notations, found for instance here). 
But again, that would be a very poor fit (notice that your distribution is skewed, i.e. not symmetrical).
edit
To get the exact result (modulo numerical/rounding errors) with Excel (or any spreadsheet editor), here is a hint http://i.imgur.com/cVwq7Tu.png (look at I22's formula). Basically, noting $Y_t = \log_2 (X_t/X_{t-1})$, you go from one column/date $t$ to the next by using the fact that $$\displaystyle P(Y_t=a) = \sum_{i \in \{-1,0,1,2\}} P(Y_{t-1}=a-i)P(a-i \rightarrow a)$$
where the transition probabilities $P(a\rightarrow b)$ do not depend on $t$ ($Y$ is a Markov chain, but you don't need to know that to understand this simple case). Then copy-paste your formula over all possible values for $t = 1 \ldots 11$ and sum all probabilities in the last column corresponding to cases where $Y_{11} \le \log_2(30)$. Takes 5 minutes all-in.
