Which activity yields the most experience?

king14nyr is correct that the Chamber of Experience - ADV is the simplest way to go, easy to overcome with just a decent friend unit, and has good experience competitive with more difficult options.

New Story Dungeons

However, now that Olderion has been released, there are new story dungeons that give higher XP/energy than the chamber. At the time of writing, these have not been added to the wiki's Dungeon Experience Chart.

You only need to spend six energy to do any of the missions in Olderion's first dungeon, Argus Plains. You won't get quite as much unit experience per energy as in the chamber, but you get much more rank experience and gil. The final mission in this dungeon, Attack of the Raptor, awards about 8200 unit experience, for 1367/energy, in addition to about 210 rank experience and about 900 gil. You can probably just auto through the whole thing if you have a friend unit with 600+ attack.

The last mission in the Shipwreck dungeon, The Charming Beast, also only requires only six energy. It awards about 12300 unit experience, for 2050/energy, and also about 230 rank experience and about 1100 gil. I have a friend Noctis with 700+ attack that can clear the mission himself, on auto. A much weaker collection of heroes could clear it if you paid attention, primarily when fighting the mermaids on the last wave.

Later dungeons require more energy per mission, resulting in less experience per energy throughout most of the rest of Olderion, if not all of it.

Metal Cactuars

Really, though, my characters make their biggest leaps in level when there is an event going that provides lots of metal cactuars, particularly metal gigantuars.

For instance, during the recent Type-0 event, you could buy up to 100 metal gigantuars from King Mog for only 100 phantoma each. Each one gives a unit 30000 xp, which is as much as a full run through the chamber. This means that, if you spend all the phantoma on gigantuars, a single run through one of the event missions nets you much more experience per energy than the chamber. The exact numbers will vary depending upon a number of factors, but I was getting 500-700 phantoma per 8-10 energy run. Doing the PRO run for 10 energy, getting 700 phantoma, and spending all of it on gigantuars nets almost 220,000xp, for about 22,000/energy.