Graph - strongly connected components

Is there any fast way to determine the size of the largest strongly connected component in a graph?

I mean, like, the obvious approach would mean determining every SCC (could be done using two DFS calls, I suppose) and then looping through them and taking the maximum.

I'm pretty sure there has to be some better approach if I only need to have the size of that component and only the largest one, but I can't think of a good solution. Any ideas?

Thanks.

Let me answer your question with another question -
How can you determine which value in a set is the largest without examining all of the values?

Firstly you could use Tarjan's algorithm which needs only one DFS instead of two. If you understand the algorithm clearly, the SCCs form a DAG and this algo finds them in the reverse topological sort order. So if you have a sense of the graph (like a visual representation) and if you know that relative big SCCs occur at end of the DAG then you could stop the algorithm once first few SCCs are found.