Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
$D^\pi\mathscr{W}(X)\cong D^b(X^\vee).$
Let's assume $X$ and $X^\vee$ are both affine, then we have a well-defined Liouville structure on $X$ which essentially doesn't depend on the compactification of $X$ by normal crossing divisors, see Seidel's work: http://arxiv.org/pdf/0704.2055v6.pdf. In particular, the wrapped Fukaya category $\mathscr{W}(X)$ is well-defined. $D^b(X^\vee)$ is the derived category of coherent algebraic sheaves.
Question Why don't we consider coherent analytic sheaves on $X^\vee$?
Since $X^\vee$ is assumed to be affine, the derived category of coherent analytic sheaves is in general not equivalent to $D^b(X^\vee)$. Recent work of Abouzaid, Auroux, etc. shows that there is an embedding
$D^b(X^\vee)\hookrightarrow D^\pi\mathscr{F}_A(X),$
where $\mathscr{F}_A(X)$ should be the partially wrapped Fukaya category introduced by Auroux in his study of Heegard-Floer homologies. This seems to suggest that we should add analytic sheaves to achieve an equivalence.
On the other hand, Seidel defined the growth rate for symplectic cohomology $SH^\ast(X)$ on the A-side
$\Gamma(X)=\overline{\lim}_\tau\frac{r(X,\tau)}{\log\tau},$
where $\tau$ is the slope of the Hamiltonian used to define $SH^\ast(X)^{<\tau}$ and $r(X,\tau)$ is the total dimension of the image of the map $SH^\ast(X)^{<\tau}\rightarrow SH^\ast(X)$ obtained by considering the continuation maps. $\Gamma(X)$ is an invariant of the Liouville structure. By the work of Seidel and Mclean, we have an upper bound:
$\Gamma(X)\leq m_X\leq\dim_\mathbb{C}(X),$
where $m_X$ is defined by looking at the boundary divisors $\overline{X}\setminus X$ in the compactification of $X$. One should be able to considering the open string analogue of Seidel's growth rate, i.e. one can analogously define the growth rate $\gamma(L)$ of the wrapped Floer cohomology $HW^\ast(L)$ of any $L\in\mathrm{Ob}\big(\mathscr{W}(X)\big)$. In his paper (http://arxiv.org/pdf/1011.2542v4.pdf), Mclean mentioned that we should expect the following upper bound:
$\gamma(L)\leq\dim_\mathbb{C}(X).$
Mirror to the growth rate $\gamma(L)$ of $L\in\mathrm{Ob}\big(\mathscr{W}(X)\big)$ we have Cornalba-Griffiths' transcendental cycle theory. This is a very old paper: http://publications.ias.edu/sites/default/files/analyticcycles.pdf. Roughly speaking, for an affine variety $X^\vee$ obtained by removing a smooth divisor from $\overline{X}^\vee$, Grauert proved that every cohomology class of $H^{2k}(X^\vee,\mathbb{Q})$ can be represented by an analytic cycle $Z$. However, such an analytic cycle $Z$ is algebraic if and only if its volume growth $n(Z,r)$ with respect to certain exhaustion $X^\vee[r]$ of $X^\vee$ remains finite. Cornalba-Griffiths refined the theory by showing that the order of the volume growth $n(Z,r)$ is related to the Hodge type of $Z$ in the compactification $\overline{X}^\vee$. More explicitly, let $\overline{Z}$ be the compactification of the analytic cycle $Z$ in $\overline{X}^\vee$, and the cohomology class $[\overline{Z}]\in H^{k+l,k-l}(\overline{X}^\vee)$ with $l\neq0$, then we have the following lower bound:
$n(Z,r)\geq C\cdot r^l.$
If one believes that the mirror symmetry between $X$ and $X^\vee$ relates non-compact exact Lagrangians in $X$ to analytic transcendental cycles in $X^\vee$, then the above discussions provide further evidences for enlarging $D^b(X^\vee)$ using analytic sheaves.