TITLE: Gödel's speed-up from constructive to classical logic?
QUESTION [11 upvotes]: Gödel's speed-up theorem implies that some proofs can get significantly shortened when allowing extra axioms.  There are concrete examples of this phenomenon for instance when moving from Peano arithmetic (PA) to PA + consistency(PA), or when moving from PA to second-order arithmetic, see for instance this question.
However, I am not aware of concrete examples of such a dramatic shortening when moving from constructive logic to classical logic. 
So, is there a known example of a statement that is both true in constructive and classical logic, that has a reasonably short proof in classical logic, but such that any proof in constructive logic would be gigantic (and thus not "human-readable")? 
Ideally, an example would be a statement that would feel "concrete enough" for the average mathematician, i.e. a statement involving numbers, graphs, algebraic structures, etc. (Friedman's examples on Kruskal's tree theorem  fit the bill)
(I would also be interested in an example that has a simple proof in classical logic, that is not yet known to be true in constructive logic but such that a proof, if it exists, must be gigantic.)
Update: There are indeed ways to construct such examples as indicated below, but which feel a bit like cheating as they essentially add the extra axiom in the statement. I am more thinking of some existence theorem of the form 
"For every object x satisfying ..., there exists an object y satisfying ...".
Think Friedman's examples, Four-Color Theorem, etc. In other words, an existence theorem where every constructive proof would be gigantic, but where there exists a short nonconstructive proof. Here again, one may hack this by artificially adding something of the form "or (A or not(A))" in the statement, but of course that's not really what I have in mind, nor the sort of statement a mathematician working with classical logic would try to prove on a regular day ;-)

REPLY [2 votes]: Here is a something which might be an example:
Sometimes there are statements $A$ and $A^*$ which a classical mathematician would consider equivalent and a constructive mathematician would not.  An example is the existence of a winning strategy for Hex.
$A(n):$ the game of Hex on an $n \times n$ board will never end in a draw AND there is no winning strategy for the second player (the first could steal it)
$A^*(n)$ There is a winning strategy for the first player in the game of Hex on an $n \times n$ board.
There is a short constructive proof of “$\forall n\,A(n)$ “
I think a constructive mathematician would accept that $A^*(13)$ is true but consider it possible that any  explicit strategy would be enormous.
A classical mathematician would easily accept “ $\forall n\, A^*(n)$ “ That might be unprovable constructively.
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A variation: 
$B$ no position can end in a draw.
$B^*$ for every position is either a winning strategy for the next player or a winning strategy for the previous player.
From Wikipedia:

In 1976, Shimon Even and Robert Tarjan proved that determining whether a position in a game of generalized Hex played on arbitrary graphs is a winning position is PSPACE-complete.[12] A strengthening of this result was proved by Reisch by reducing quantified Boolean formula in conjunctive normal form to Hex played on arbitrary planar graphs.[13] In computational complexity theory, it is widely conjectured that PSPACE-complete problems cannot be solved with efficient (polynomial time) algorithms. This result limits the efficiency of the best possible algorithms when considering arbitrary positions on boards of unbounded size, but it doesn't rule out the possibility of a simple winning strategy for the initial position (on boards of unbounded size), or a simple winning strategy for all positions on a board of a particular size.