Everywhere I see on the internet they say that discrete probability distribution functions have a countable number of outcomes, and continuous have uncountable infinite number of outcomes.
However if your domain is infinite dimensional with finite number of elements in each dimension, then clearly there is uncountable infinite many outcomes but discrete.
An example is a single experiment of flipping a coin infinite number of times.
So what am I missing?
The coin flipping space is not a discrete probability space – each outcome has probability zero.
For any $n$, only a finite number of outcomes (indeed $n$) with probability $\ge 1/n$ is possible. Since any outcome with a positive probability must have probability $\ge 1/n$ for some natural number $n$, only a countable number of outcomes can have positive probability.
(Note: By outcome I mean a single point in the probability space. Contrast this with events, which are subsets of the space.)
We sometimes consider experiments that can involve an unbounded number of coin tosses, but rarely does anyone consider an infinite number of coin tosses.
You could create a probability distribution over the infinite power set $\{0, 1\}^\aleph$ where $0$ and $1$ represent heads and tails. You could then assign non-zero probabilities to a countable number of individual elements of this power set, but then you would have something that was essentially a discrete distribution over a countable set (with an uncountable number of "outcomes" that all together have zero probability).
Alternatively, you could assign non-zero probabilities to certain subsets of the power set. It's not clear to me how to assign non-zero probabilities to more than a countable number of disjoint subsets. If you can do so, we still have to ask whether we want to consider such a thing to be a discrete distribution.
There are distributions that are neither continuous nor discrete. It's not an either-or thing.
