Why use Azure to extend Dynamics 365?

What are some good use cases for building logic for your Dynamics 365 using Azure?

1. Inconsistent workload demands
Have you started using the Dynamics 365 portal? It is pretty cool, but if you have waves of people hitting the web site and making updates, and you have numerous plugins and workflows running, you may find that your system can be sluggish if not outright slow. By moving some of your plugins and workflows to an Azure service, you can mitigate some of the effects of the portal. Consider using Azure Functions.

2. Offloading batch processes
Are you an association that needs to update 100k membership records at the same time, or generate 20k invoices on the last day of the month? You shouldn't even try to do that with workflows or your users coming after you with torches and pitchforks. Just like #1 above, you could have an on-demand service running in Azure to do the dirty work without seriously impacting your users. Consider using a Cloud Service that you can deploy, run, then shut down.

3. Scheduled jobs are not easily managed within D365
You want a system update to run at 5PM every Friday? I dont think you can do it in D365 without writing some plugin code and a workflow, but Azure has an OOTB solution for that. Take a look at Azure Logic Apps. You can schedule them to run at a specific time.

4. You are moving to D365 from an on-premises environment that uses SQL Stored Procedures
You have some SQL queries and views you use to support reporting systems. CRM has a feature to replicate an entire entity to an Azure SQL database. You want to push updates back? That is a bit more tricky, but perhaps you can manage it with Azure Logic Apps or Azure Functions.

5. Plugins and workflows in D365 have a timeout limit of 2 minutes
Yep, trying to update more than a few records in a single plugin is detrimental to your employment. In one case, I found that a client had created a bunch of small plugins, but what the developer(s) didn't realize is that they had created a chain of plugins that was over 1500 lines of code and periodically failed from time to time, depending on the current server load. The worst part was that there was no way to re-trigger the plugins because they were designed to work only with a Create event message, and the code did not support idempotent transactions, so you could only run it once and hope it worked the first time. Consider using an Azure Cloud Service, Web Job, Service Fabric, or Function, depending on what you are trying to accomplish.

(update)
6. Consolidate Business Logic 
Perhaps you have several places in your system that do the same thing, like processing a credit card. If you put the logic in a plugin, it is not easily accessible to your web site (yes you can do it, but it is not a good user experience). You could use a WebJob that works with both CRM and your web site.