Using Certificate Transparency to monitor your organization's TLS activity
One of the obvious things that you can do with Certificate Transparency is to monitor the CT logs for bad people somehow getting a certificate for one of your websites. If you're Paypal or Google or the University of Toronto, and you see a CT log entry for a 'www.utoronto.ca' certificate that isn't yours, you can ring lots of alarms. You can do this with actual infrastructure (perhaps based on the actual logs, and see also, and also), or you can do this on a manual and ad-hoc basis through one of the websites that let you query the CT logs, such as Google's or crt.sh, or maybe Facebook's (which of course requires a Facebook login because Facebook, that's why).
But there's another use for it, and that is looking for people in your own organization who are getting properly issued certificates. Perhaps I'm biased by working in a university, but around here there's no central point that really controls TLS certificates; if you can persuade a TLS certificate provider to give you a certificate, people will. And these days, the existence of Let's Encrypt means that if you have control over your own hosts, you can probably get certificates for them. If you are in such an organization, monitoring Certificate Transparency logs is one way to keep track of who is doing roughly what with TLS, perhaps discover interesting services you want to know about, and so on.
(Perhaps you are saying 'we control who gets to run TLS services because we control the perimeter firewall'. Do you control DNS too, so that people can't point off to things they're hosting in AWS? You probably don't want to go that far, by the way, because the alternative is for people to buy their own domain names too and then they won't even show up in your CT monitoring.)
You don't even have to be at the top of an organization to find this interesting, because sometimes there are subgroups all the way down. Some of our graduate students run machines that can be reached from the outside world, and I'm sure that sooner or later some of them will want a TLS certificate and discover Let's Encrypt. It's reassuring to know that when this happens we have at least some chance of finding out about it.
(Not an entirely great chance, because sometimes professors set up new domain names for graduate student projects and don't tell us about them.)
PS: Of course, as a bystander in your (overall) organization you can also use CT logs to satisfy your curiosity about things like how common Let's Encrypt certificates are, and how broadly spread they are across your organization. Is your group one of the few areas actively experimenting with them, or are a whole lot of people using them all over the place?
(All of this is probably pretty obvious, but I feel like writing it down.)