The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on "something" (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).