In the ordinal-numbers model that your notation leans toward, 2+2+2+... and 2+1+2+1+... are definitely the same infinity, both ω.
On the other hand, you reminded me that conditionally convergent series may converge to different sums depending on how you choose to group their terms, which is fun. (That is, [(s_1 + s_2) + s_3 + (s_4 + s_5) + s_6 + ...] may have a different sum than [(s_1 + s_2) + (s_3 + s_4) + (s_5 + s_6) + ...] does.)†
† I hope this is true; I can't find a cite for it. The more famous theorem is that you can get any sum you want by reordering the terms, but I think regrouping is good enough to get you different sums, if not arbitrary sums.
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Thinking about it more, the idea would be that the sum of a series is of course the limit of the sequence of partial sums. By artfully parenthesizing the terms of an original series, we can reduce it to a new series with a different sequence of partial sums. This sequence must be a subsequence of the original sequence -- parenthesizing the series essentially just means ignoring certain partial sums.
So then the question, if we're willing to start waving our hands, is "when (infinite) sequence S has limit L, and (infinite) sequence S' is a subsequence of S, must S' also have limit L?".
The infinitude of S' is going to require the answer to be yes, so regrouping will end up not being good enough to change the sum of a conditionally convergent series. Sorry. :(
However, if a series does not converge or diverge, such that its sequence of partial sums has multiple limit points, artfully parenthesizing the terms of that series should be able to get you a new series that converges on any limit point of the old one.