My concierge was already listed in this thread as a useless one.
However I find my concierge very helpful. The main benefit is that I can transact most business by email. If there's any kind of billing or reservation problem, I just shoot off an email and then rest assured, knowing that it's solved.
Example: I recently applied a free night to a stay, but the hotel charged me for all the nights. It literally took 15 total seconds to ask the concierge to fix this.
The concierge won't break or even bend any rules for me, which is exactly how I prefer it. I get all bent out of shape when companies or people don't follow their own rules, therefore I try to follow all the rules as written. Still, it's permissible to ask for a special request on occasion. The worst that can happen is they say "no", which is not really a loss. If I have a special request and the concierge won't help, I simply phone the hotel directly (or ask the concierge to do it on my behalf).
Example: recently I had an 8-night stay and wanted to use a TSU for just 7 of the nights. Hyatt policy is that you must use TSU to cover _all_ of your stay. So it would have cost me two TSUs for the 8-night stay. I wasn't willing to blow a TSU for a single night. I wanted to stay 7 nights with a TSU and 1 night without. Concierge would not budge on this issue (which is understandable, because concierge is there to help with reservations that follow Hyatt's written policy). So I called the hotel and asked them to make an exception. Hotel agreed to put me in a regular room for one night and a suite for 7 nights.
Overall I'd say that the concierge service is a huge benefit of being a Globalist.