A near-maximum insurance claim is one falling within a distance $a$ of the current maximal claim. In this paper, we investigate asymptotic behavior of normalized numbers of near-maximum insurance claims under the assumption that the sequence of successive claim sizes forms a strictly stationary process. We present the results in a general form expressing limiting properties of normalized numbers of insurance claims that are in a left neighborhood of the $m_n$th largest claim, where $m_n/n$ tends to zero and $n$ is the number of registered claims. We also give corollaries for sums of near-maximum insurance claims.