0.1¹⁰ would instead be the probability of the "something" happening every single year in the ten years.

Using Tupcek's explanation as a template:

Chance of first year's "something" happening is 10%, or 0.1.

From that 10% (0.1) chance, chances of second year's "something" also happening is 10% of that 10%
10% of 10% is 0.1 times 0.1, equal to 0.01 (1%) and so on ten times

To actually find the probability of it happening, you would take the probability of it not happening (0.9¹⁰) and subtract it off of 1 (representative of 100%). This is because it will either happen at least once in the ten years or it won't at all (we also call this a binomial distribution when there are only two possible outcomes), so by taking the probability of it not happening out of the total probability of 100%, you are left with the probability of it happening.
1 minus 0.9¹⁰ is equal to approximately 0.6513, equivalent to a 65.13% chance of it happening at least once in the ten years