Your question may be recast as: 

Let $A$ be an integral symplectic matrix $A\in Sp(n,\mathbb{Z})$. Is it true that there exists $N(n)$ such that there is $k\leq N(n)$ with $tr(A^k)>2 $ (with a few small exceptional cases)?

In fact, one may ask a similar question for $A\in SL(n,\mathbb{Z})$? 

Let the singular values of $A$ be denoted $\lambda_1,\ldots, \lambda_n$. Then $tr(A^k)=\sum_{k=1}^n \lambda^k =p_k(\lambda_1,\ldots,\lambda_n)$, where $p_k$ is the [power symmetric polynomial][1] of degree $k$ in $n$ variables. Now, we observe that $p_k$ is a polynomial in $p_1,\ldots, p_n$ for $k>n$. I found [another mathoverflow question which computed the first few of these][2]. 

Consider $n=2$ (note, $SL(2)=Sp(2)$), and suppose $p_i(\lambda_1,\lambda_2) \leq 2, 1\leq i\leq 2$. It turns out that if $-2\leq p_1 \leq 2$, then $|tr(A^k)|\leq 2$ for all $k$, and $A$ is either parabolic or elliptic (and similarly if $|p_2|\leq 2$). So assume $p_1,p_2 \leq -3$. Then $p_3=-p_1^3 +\frac32 p_1p_2 \geq \frac{81}{2}$. Thus, $N(2)$ exists.

Now, consider $n=3$. Again, assume $|p_i|\leq 2$ for $1\leq i \leq 3$. There are only finitely many possibilities for the characteristic polynomial $det(xI-A)$ in this case, since the elementary symmetric polynomials are functions of the power symmetric polynomials. So there is some definite power $N(3)$ which will have $tr(A^k)>3$ for $k\leq N(3)$ in this case. Thus, we may assume $p_i \leq -3$, $1\leq i\leq 3$. Then $p_4=\frac16 p_1^4−p_1^2p_2+\frac12 p_2^2+\frac43 p_1p_3 \geq \frac16 3^4 + 3^3+\frac12 3^2 +\frac43 3^2 = 57.$ Thus, $N(3)$ exists.

Similarly, for $n=4$,  whenever $p_1,\ldots, p_n \leq -3$, $p_5 \geq 102$, so a similar argument shows $N(4)$ exists. 

I'll make the bold conjecture that this pattern holds more generally, based on this meager evidence. Thus, $p_{n+1}$ should be a polynomial of $p_1,\ldots, p_n$ (for $n$-variable power symmetric polynomials) which is $>2$ when $p_1,\ldots, p_n\leq -3$. This conjecture would imply a positive answer to your question. 

  [1]: http://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial
  [2]: http://mathoverflow.net/a/37235/1345