Risk and probability: Is there always infinite risk?

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If there are an infinite number of events with a cost (risks), and total Risk is calculated by the sum of all individual risks multiplied by their probability, does everything have an infinite risk associated with it?

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No. If the events $a_n$ all have a risk of $1$, and a probability of $2^{-n}$, then the total risk is:

$$\mathrm{risk} = \sum_{n=1}^\infty \operatorname{risk}(a_n)\Pr(a_n)=\sum_{n=1}^\infty 1\cdot 2^{-n} = 1$$