The Peloponnese Region of Greece is an archaeologists dream location. Already inscribed with 5 locations of great importance, this location equals those. On the tentative list since 2014 it still holds its weight amongst these sites and should be on your visit list. I am not sure why it has not been inscribed to date.
I was contemplating not even visiting, however had a spare day with the hire car and rather than tick off more tentative Medieval Bastioned Fortifications of Greece or Ancient Theatres of Greece (visited a few now) we visited this singular location as a side trip as we were travelling up from Sparti to Patras to ferry out of Greece. I am so glad we did; it is seriously underrated.
15 Euro entry fee to visit, which is middle of the road for prices across Greece to visit these sites, however you do get your money's worth at this site with the amount of ruins and large footprint across the protected area. There is no 'museum' part of the complex, the 'museum' is the ruins themselves with a variety of information boards explaining the site and area. It is a dog friendly location, and we were only asked if our dog was vaccinated, which is a first for us. We spent about 3 hours on the site itself and involved a fair bit of walking and exploring, so if visiting, make sure you take some water with you.
With a lot of the archaeological sites I have visited, there is a large degree of imagination needed to interpret the site, obviously aided by information boards, however Messene has considerable structures and ruins in place that less imagination is needed as there are substantial ruins still in place to admire and observe. I was particularly impressed with the large stadium, gymnasium and Palaistra which are at the bottom of the hill as you meander down the archaeological site. This does not detract from the ruins of the Theatre, Agora or Asklepion. The photo supplied is one of the Funerary Monuments which is located at the end of the Stadium as most photos supplied show the stadium (as it is impressive).
We visited on a Wednesday in November arriving at about 11.00am and the site had less than 10 people present. As we were leaving a number of tour buses at arrived involving a number of tour groups and school groups which brought the numbers up to about a 100 on site, but still wasn't that busy that it ruined your visitation experience or the serenity of the site. It is a definite thumbs up from me and hopefully it will only be a matter of time before it receives its permanent inscription.