What admin access does connecting an integration need?

It differs per integration.

Each person connects their own Google Calendar, Office 365 (Outlook) and Slack from My Integrations, while Tempo and Microsoft 365 are global and configured once by an app admin under App Settings > Global Integrations; Slack additionally needs a Slack workspace administrator to install the app into the workspace.

Permissions and Access Requirements - where this is covered in full.
User Permissions - more questions on this topic.

What admin access does connecting an integration need?

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