Hook priority in admin with custom plugin

The codex entry you site is simply wrong! There are a lot of hooks fired before admin_init during admin requests, not least of which is init, which a lot of plugins use to do their initialization.

In particular, the TGM Plugin Activation plugin hooks into init (with the default priority = 10) with a function that calls do_action ('tgmpa_register').

Hence, you’ll need to do something like:

// make sure we register this hook func with lower priority value than 
// TGM Plugin Activation does
add_action ('init', 'sr_admin_init', 9) ;

function
sr_admin_init ()
{
    if (!is_admin ()) {
        return ;
        }

    // I NEED THIS TO TRIGGER FIRST
    wp_die("sr_admin_init");
}

Edit

To be more charitable to the author of the codex entry you site: admin_init is the 1st “strictly admin”-related hook fired for admin requests. It’s just not the 1st hook fired.