The choice of variables depends on some other criterion you're trying to optimize. For example, if your goal is to make the best possible prediction of GDP given the predictors (best = unbiased with smallest standard error), then including all the predictors would be the way to go. Predictors can't ever add noise (if they were totally unrelated to the outcome their coefficients would be zero), so in a very basic sense more are better.
But we're often asking a slightly different question, such as "what is the best prediction I can get using only K predictors" or "What is the optimal adjusted R^2 value I can obtain?" (Adjusted R^2 starts with the usual R^2 and then "penalizes" for the number of predictors).
The general process you're engaged in is "model selection." There are various algorithms for this (stepwise regression, best subsets regression, etc.) Stepwise has fallen out of favor (see http://en.wikipedia.org/wiki/Stepwise_regression) , and for a relatively small number of predictors a best-subsets algorithm would give you the an optimal tradeoff between model parsimony and predictive power. But if you want to try stepwise then, yes, forward stepwise does what you propose: pick the predictor with the highest R^2, generate the residual from that Y=x1 + e1 model, then pick x2 from the remaining predictors that best explains e1 (e1 = X2 + e2), rinse and repeat.
But again, you might be going for a completely different criterion, such as a model that best explains the processes that actually cause (lead to) GDP. There is a distinction between "explanation" and "prediction." The former would rely on some substantive understanding about what the variables actually mean and how they function in an economy - the latter is just about variables (I as a non-economist can answer your questions about prediction, but not about what variables best EXPLAIN GDP)
Bottom line - the answer to your question depends on a careful consideration of the criteria you're trying to optimize.