elementary set theory - Cardinality of the set of at most countable subsets of the real line?

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with different language: What is the cardinality of at most countable subsets of $\mathbb{C}$ (or $\mathbb{R}$, if you prefer)?

I asked my advisor and he surprisingly wasn't sure, though he suspects that the set of subsets in question has a larger cardinality than $\mathbb{R}$.

Thanks a lot!

Edit: Certainly if we only consider finite subsets, then this set of subsets has cardinality equal to $\mathbb{R}$.

Edit2: Realized my wording was wrong. I'm actually looking for the cardinality of the set of sequences with entries in $\mathbb{C}$, not the cardinality of the set of at most countable subsets of $\mathbb{C}$. However, both questions are answered below, and both turn out to be $|\mathbb{R}|$.