Clustering is a fundamental primitive in manifold applications. In order to achieve valuable results in exploratory clustering analyses, parameters of the clustering algorithm have to be set appropriately, which is a tremendous pitfall. We observe multiple challenges for large-scale exploration processes. On the one hand, they require specific methods to efficiently explore large parameter search spaces. On the other hand, they often exhibit large runtimes, in particular when large datasets are analyzed using clustering algorithms with super-polynomial runtimes, which repeatedly need to be executed within exploratory clustering analyses. We address these challenges as follows: First, we present LOG-Means and show that it provides estimates for the number of clusters in sublinear time regarding the defined search space, i.e., provably requiring less executions of a clustering algorithm than existing methods. Second, we demonstrate how to exploit fundamental characteristics of exploratory clustering analyses in order to significantly accelerate the (repetitive) execution of clustering algorithms on large datasets. Third, we show how these challenges can be tackled at the same time. To the best of our knowledge, this is the first work which simultaneously addresses the above-mentioned challenges. In our comprehensive evaluation, we unveil that our proposed methods significantly outperform state-of-the-art methods, thus especially supporting novice analysts for exploratory clustering analyses in large-scale exploration processes.