In the vSphere Client and vSphere Web Client you can find how many days your host has been running. But what happened before that period? How often did your host boot in the past? This information is stored in the file /var/log/vmksummary.log on your ESXi-host. The file is updated hourly with information how long your host has been up and how many virtual machines are running at that moment.
In the image below you can see that my host was running for 68 days and at that time I have reboot the host, which is also written into this file.
So when we use this information and a simple Linux-command we can find all the entries in this file that tell us when the host has been booted:
grep booted /var/log/vmksummary.log
As you can see in the image below the host has been booted 9 times since December 28, 2013.
Note: This information is only available if your host has persistent storage for the log files. When your host is booting from a USB-disk for example and there is no permanent scratch partition then this information does not survive a reboot. How to configure a persistent scratch partition is explained in this VMware Knowledge Base article: Creating a persistent scratch location for ESXi 4.x and 5.x (1033696)