As far as I'm aware, there is not a way to show the full comment history in email notifications in Service Desk without additional plugins.
One plugin that can provide that functionality is Email This Issue.
Here's another question asking about this, where the top answer shows the template you can use to include all comments in emails:
I looked around and found several feature requests for this issue. Here's the one that seems to be the original - have a look and watch/vote on the issue: https://jira.atlassian.com/browse/JSDSERVER-2044
Development on Jira Service Desk is still happening at a pretty fast clip. I don't work for Atlassian, but I can say that if that issue gets a lot of votes, there's a good chance of someone on the Service Desk team taking a look at it.
Hello Atlassian Community! I'm Teresa, the Product Marketing Manager for Jira Service Desk Server at Atlassian. Curious about the latest updates in the Jira Service Desk Server...
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