  We construct a new analogue of the BGG category $\mathcal O$ for the
infinite-dimensional Lie algebras
$\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main
difference with the categories studied in \cite{Nam} and \cite{CP} is that all
objects of our category satisfy the large annihilator condition introduced in
\cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of
$\fg$ are not conjugate, one can eliminate the dependency on the choice of
$\fb$ and introduce a universal highest weight category $\mathcal {OLA}$ of
$\fg$-modules, the letters $\mathcal{LA}$ coming from "large annihilator". The
subcategory of integrable objects in $\mathcal {OLA}$ is precisely the category
$\mathbb T_{\fg}$ studied in \cite{DPS}. We investigate the structure of
$\mathcal {OLA}$, and in particular compute the multiplicities of simple
objects in standard objects and the multiplicities of standard objects in
indecomposable injectives.
