How do I get the last available hook_update_N() for a module? I'm having trouble getting a custom module to install and update successfully.  I'd like to examine the database for the modules update_N numbers. Is that possible? If it is, where do I look for it?
They're in key/value storage, which in the database would be the key_value table. Look for entries with a collection of system.schema, and a name matching the machine name of the module you're interested in.

Through the API, it's a bit easier:

$current_version = drupal_get_installed_schema_version('name_of_module');