In short, not on the devices themselves. However, there are ways to make it harder for students to delete their entire browsing history, and make sure that browsing history is captured somewhere in addition to the devices themselves.
The easiest way to control student browsing (and which tabs students have access to) is on Chromebooks, which were designed to be managed and deployed across entire school districts. However, you can still manage browsing on any device through browser-specific steps.
Within the Google Admin console, there is no one specific setting to choose that will keep students from deleting their browser history. Some admins have used workarounds, such as blacklisting the following sites: chrome://history, chrome://history-frame, chrome://settings-frame/clearBrowserData, and history.google.com/history/ in the Admin Console so students can’t access their history page. A huge disadvantage to this is that in addition to it being unreliable, in order to be able to view their history, you would have to un-restrict them.
So, what can we do?
The only way to ensure that students can’t permanently delete their Chrome history is to make sure that it is captured with a 3rd party tool who keeps a running history. There are a number of Chrome extensions available that can ensure that even if the students do delete their history, that the history is recorded externally. For example, companies like GoGuardian, and Securly offer ways for admins to monitor their students’ Chromebook usage, as well as be able to see a detailed history of each site the student visited. GoGuardian for Admins offers other features as well, such as the ability to filter school devices when they go home, custom profiles in order to manage student chromebook usage per OU (which can sync with your Google Admin Console), YouTube history, an extensive YouTube filtering tool, Google Doc history, Search History, flagged activity reporting, and many others.
**Update: Since writing this article, as of April 2017, there is now a way to block the deleting of history within the Google Admin Console.**