First, let us generate a large sum with terms similar to your mini-example.

hugeExpr[n_] := 
  Plus @@ Array[
    RandomInteger[{-10, 10}] x^
       RandomInteger[{-3, 1000}] Sqrt[x + RandomInteger[{-10, 10}]]^
       RandomInteger[{-1, 1}] &, n];

expression = hugeExpr[300000];

The actual number of addends (Length[expression]) might not be 300K, but it will still be a large count. Now, the answer to your problem.

ParallelMap[Series[#, {x, 0, 2}] &, expression, Method -> "ItemsPerEvaluation" -> 15]

I tried it on a expression with over 40K terms and it took less than 2 seconds. If your expression is orders of magnitude larger than that, I would suggest you split it in chunks of say 100K addends, use ParallelMapon each chunk separately, save those results to individual files, and add them up later.

First, let us generate a large sum with terms similar to your mini-example.

hugeExpr[n_] := 
  Plus @@ Array[
    RandomInteger[{-10, 10}] x^
       RandomInteger[{-3, 1000}] Sqrt[x + RandomInteger[{-10, 10}]]^
       RandomInteger[{-1, 1}] &, n];

expression = hugeExpr[300000];

The actual number of addends (Length[expression]) might not be 300K, but it will still be a large count. Now, the answer to your problem.

ParallelMap[Series[#, {x, 0, 2}] &, expression, Method -> "ItemsPerEvaluation" -> 15]

I tried it on a expression with over 40K terms and it took less than 2 seconds. If your expression is orders of magnitude larger than that, I would suggest you split it in chunks of say 100K addends, use ParallelMap on each chunk separately, save those results to individual files, and add them up later.