Let $X_1, \ldots $ be a sequence of $IID$ Bernoulli(1/2) random variables, valued in $-1$ or $1$. Let $\Omega = \{-1,1\}^{\mathbb{N}}$ be the corresponding path space.
In order to expand my intuition, I'm trying to find some interesting events $E$ inside of $\Omega$ that have $0 < P(E) < 1$.
Let me qualify: $X_1 = 1$ is not an "interesting" event. Nor is any other rectangle, or artificial union or intersection of rectangles.
An interesting event would be something like : $\{ \Sigma_{i = 1}^n X_i \geq n $ for infinitely many n$\}$, or $lim X_i = 1$. Unfortunately, these events are exchangeable or tail respectively, so by 0-1 laws they have probability $0$ or $1$.
Perhaps "interesting" means defined by a formula or property about the sequence. I'm not sure what I mean exactly, so this is a "soft question."
There are certain structures you can impose that are neither exchangeable or tail, but they also force probability zero or one. For example, $X_1 > X_2 > \ldots$ is measure zero.
By identifying $\Omega$ with $[0,1]$ Lebesgue, one can carry over interesting sets (fat Cantor sets, or something), but I don't see any with natural descriptions.
So this makes me wonder, as a follow up question -- is there a zero one law for IID bernoulli's that works under some very general assumptions about the logic structure of the defining formula? Or am I just not being creative enough? This reminds me of the story about the search for "random" binary sequence (Martin-Lof randomness, Kolmogorov complexity, collectives, etc.)
As you note with your examples, it's easy to fall into the trap of making everything a tail event. You have to have an event that depends on a large number of them. Here's a way to get a whole bunch of related questions. We know (by Kolmogorov's three-series test) that $$S:= \sum \frac{X_k}{k} $$ converges almost surely. There are a whole bunch of non $0$-$1$ events related to $S$; in parcular, $\mathbb{P}(S > 0 ) = 1/2$.
You can use $S$ to get pretty crazy stuff. It can be shown that $S$ is absolutely continuous w.r.t. Lebesgue measure with density supported on all of $\mathbb{R}$. This means that you can use it to generate any continuous random variable with a density by using a Radon-Nikodym derivative. You could also generate countably many independent copies of $S$, since you can embed $\{0,1\}^{\mathbb{N}\times \mathbb{N}}$ in $\{0,1\}^\mathbb{N}$, thereby giving countably many, let's say, standard normals. Countably many normals is enough to generate Brownian motion on $[0,1]$ (usually by adding standard normal multiples of tent functions); doing this countably many times gives Brownian motion on $\mathbb{R_{\geq 0}}$. Here, you can certainly ask many interesting questions.
Basically the whole point is that you can actually get a lot of non-trivial events out of these; the hard part is getting to a continuous random variable.