Let $X_1$ and $X_2$ be two cones over the hawaiian earring and let $X$ be the wedge sum of $X_1$ and $X_2$ (of course you join them in the special point of the hawaiian earring). How do you prove that $X$ is not contractible? Thanks
See the proof of Theorem 2.6 of
A copy of this paper is available on the second author's webpage here.
The space you are talking about is sometimes called the Griffiths space. As Henry Horton suggests, to prove it is not contractible you can show the fundamental group is non-trivial (even though its Cech homotopy groups are trivial). The reference already given is a good one though I'll add that Griffith was the first to show this in
H. B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. 5 (1954), 175-190.
Another discussion on non-contractible one-point unions of simply connected spaces is in
K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proc. Amer. Math. Soc. 116 No. 1 (1992) 239-249.