<<Sharing this information for broader community usage>>
@Peter Thurwachter (MINDTREE LIMITED) - Currently Custom Application Insights metrics with no dimensions are free. If you are sending a generic azure monitor custom metrics, there is cost.
Also, collection of custom metrics dimensions is turned off by default because in the future storing custom metrics with dimensions will be billed separately from Application Insights, while storing the non-dimensional custom metrics will remain free (up to a quota). You can learn about the upcoming pricing model changes on our official pricing page.
Additional documentation reference: Log-based and pre-aggregated metrics in Azure Application Insights - Azure Monitor | Microsoft Learn
Hope the above information helps, let us know if you have any further queries.