This new historical survey may interest some: Open-Access Repositories Worldwide, 2005–2012: past growth, current characteristics, and future possibilities…

“This paper reviews the worldwide growth of open-access (OA) repositories, 2005 to 2012, using data collected by the OpenDOAR project. Initial repository development was focused on North America, Western Europe, and Australasia, particularly the United States, United Kingdom, Germany, and Australia, followed by Japan. Since 2010, there has been repository growth in East Asia, South America, and Eastern Europe, especially in Taiwan, Brazil, and Poland. During the period, some countries, including France, Italy, and Spain, have maintained steady growth, whereas other countries, notably China and Russia, have experienced limited growth. Globally, repositories are predominantly institutional, multidisciplinary and English-language based. They typically use open-source OAI-compliant software but have immature licensing arrangements. Although the size of repositories is difficult to assess accurately, available data indicate that a small number of large repositories and a large number of small repositories make up the repository landscape.”

I wondered if this also discussed “growth” in terms of “the growth in indexing”. But sadly the article is behind a Wiley paywall (Update: also self-archived). The poor state of repository indexing by Google, and the probable reasons for it, are however addressed in this 2012 paper from the University of Utah: Invisible Institutional Repositories: addressing the low indexing ratios of IRs in Google Scholar.