Example of a probability theorem that requires axioms in addition to Kolmorogov's? Probability theory, in it's more general form, is axiomatized by Kolmorogov's axioms:
Kolmorogov's Probability Axioms
Let $(\Omega,\mathcal{F},P)$ be a measure space.
The three axioms are:


*

*Positivity: $E \in \mathcal{F} \implies P(E)\geq 0$

*Unit Measure of $\Omega$: $P(\Omega)=1$

*$\sigma-$additivity: Let $E_i$ be a countable sequence of disjoint events, then:
$$P\left(\biguplus_{i=1}^{\infty} E_i\right) = \sum_{i=1}^{\infty} P(E_i)$$



Now, all theorems in probability theory should be consistent with these axioms. However, it that doesn't imply that they should be derivable from these axioms. 
My question relates to how much of probability theory we can reconstruct from just these axioms? Do we hit a theoretical wall that requires new axioms to allow for further theoretical development in a particular sub-field?
This may seem obvious to those who study this for a living, but as a practitioner (read: not a theoretician), it is not obvious if these axioms are sufficient to reconstruct modern probability theory without specialized axioms for particular sub-fields (e.g., stochastic processes, large deviations, random graphs, percolations, etc.). 
I'm not saying that these sub-fields don't have their own definitions and concepts, a simple google search quickly shows this. I am asking the deeper question of whether these additional assumptions/axioms are more akin to "convenience axioms" (which are just re-casts/reformulations of pre-existing axioms in a new terminology for the sake of consistency) or are they truly independent of (meaning not derivative or equivalent to) the above axioms?
In response to the close requests
I see that some have voted to close because its not specific enough. I would like to get an answer to this, so I've given some thought about how to make this more specific. See my more restricted question below:
Restricted/Revised Question
I will accept as an answer one of two things:


*

*An example (read: one) of a theorem in probability that relies upon axioms in addition to those above.

*An explanation as to why no such theorems exist.

 A: When dealing with collections of random variables indexed by time, we need more than just a probability space $(Ω, \mathcal{F},P)$ to capture the notion of 'information knowable at time $t$'. It is termed a filtered probability space, which is a 4-tuple $(Ω,\mathcal{F},(\mathcal{F_t})_{t\geq 0},P)$ where the new ingredient is a filtration, i.e. a collection of sub-$\sigma$-algebras $\mathcal{F_t} ⊂ \mathcal{F}$ such that
$$ s<t \implies \mathcal{F_s} ⊂ \mathcal{F_t}$$
This normally comes with the concept of an adapted stochastic process $(X_t)_{t\geq 0}$ i.e. a collection of random variables where $X_t$ are $\mathcal{F_t}$-measurable.
Why do we want to have collections of $\sigma$-algebras? Well, it turns out that conditional expectations (and thus, conditional probabilities) are most naturally expressed in terms of sub-$\sigma$-algebras. The game of guessing a result in the future based on information in the present is therefore easy to discuss in such a setting. 
For example, the statement that your winnings $X_n$ from playing in a casino will on average decrease as you play, can be formalised as saying that $X_n$ forms a (discrete-time) supermartingale, i.e. if $k<n$, then
$$\Bbb E[X_n | \mathcal{F_k}] \leq X_k $$
For another example, the markov property is that
$$ P(X_t ∈ A|\mathcal{F_s}) =  P(X_t ∈ A|X_s)$$
A theorem that one soon encounters after these definitions is the Optional Stopping Theorem. For a celebrated result that requires this framework, I refer you to
the Black-Scholes Model.
