An Aesop: There's the classic "a person's a person no matter how small" as well as a celebration of creative thinking and "thinkers" that runs as the undercurrent of the whole show. The Cat even begins the final number by asking the audience what they think happened after the story.
Arc Welding: Arguably. It puts a lot of Dr. Seuss' many many stories into a single continuous setting, and a good number of them figure into the main plot. See the Continuity Porn entry below.
Beautiful All Along: Gertrude, although she comes to realize this herself, and for practical reasons as well.
Big Bad Ensemble: The Sour Kangaroo for the Jungle of Nool and General Genghis Kahn Schmitz for Who Ville, though Schmitz is the Big Bad from Jojo's perspective only. Regardless, both Schmitz and the Kangaroo pull Heel-Face Turn 's by the end of the story.
Breaking the Fourth Wall: The Cat regularly does this silently, but his last line before the final song is addressed right at the audience "And then guess what happened? Well? What do you think?"
Continuity Porn: Of Dr. Seuss's stories. The plot of the musical tries to mix as many of them into one coherent story, and takes a few liberties in order to do so, and has a somewhat complex structure as a result. However, a lot of them figure in as Shout Outs and are more or less insulated from the main plot, therefore remaining unchanged other than their setting. The Other Wiki has an impressive list of all the stories that cameo in the book.
Crowd Song: "Oh, the Thinks You Can Think", among others.
Cut Song: The original Broadway version and CD have a song called "A Day For The Cat in the Hat," in which the Cat tries to entertain Jojo while his parents are attending a conference with his teacher. The song was cut for the touring production, which also featured a rewritten script that made the Cat JoJo's guide throughout the whole play.
Just to give you an idea, while looking for Horton, she:
Slogged through some swamp
Was attacked by bees and dogs
Trudged through snow in subzero conditions
Tripped and fell off of a hill and landed on a jagged shoal (spraining her toe in the process).
Was ran over.
At some point during all this, she took seven weeks (almost two whole months) and found the clover with the speck of dust that holds the Whos' world. Keep in mind that this was in a field of clovers that were all identical. And she herself cannot hear the Whos and so had to just look for a speck, unlike Horton, who could listen for them.
Some think that the plight of the Who's, who are too small to represent themselves, is representative of a position on abortion (taking Word of God ('s wife) into account, this may be Misaimed Fandom), or of underrepresented labor workers. The idea itself could probably be fitted into any number of political agenda's, just as with the original Horton Hears a Who!.
Actually, they're the Japanese. Theodore Geisel (ie Dr Seuss) used to be a politcial cartoonist for PM during the war and made some terribly racist cartoons towards the Japanese. After the war, he went to Japan, made friends with some locals and had some Creator Backlash towards his previous work and wrote Horton Hears a Who!!. It serves as an allegory of Hiroshima and the post-war occupation.
Pet the Dog: General Schmitz is given one in Solla Sollew, where he joins in on singing about the mythical place for a line or so, depending on production. He also panics when Jojo is leaving because Schmitz was to horrible to him, because he realizes too late that Jojo is going into a minefield. When he thinks Jojo is dead he insists the boy is hero, instead of a deserter, and returns his sword and hat to his parents respectfully and with regret.
Shout-Out: One line in "Biggest Blame Fool" is "Acting as if he's holding a jewel", referring to how dumb the other animals think Horton the elephant is by believing that a small speck of dust can talk, even though he's the only one who hears it. There was a movie called The Magic Adventures Of Mumfie which had an elephant who had to protect a magical jewel belonging to a queen.
They've Come So Far Song: Seussical either has one such song, or has multiple songs that apply to different characters. "All for You" is one, relating the sheer hell Gertrude has gone through to get to Horton. However, everything mentioned in the song happened off stage.
Villain Song: "The Military Academy" for Schmitz. And, if you include Mayzie as a villain, "Amayzing Mayzie".
"Biggest Blame Fool" for the Sour Kangaroo, and "Monkey Around" for the Wickersham Bros.
You Are Not Alone: Both Horton and JoJo are feeling ostracized and lonely near the beginning of the show, then they manage to talk to each other and finally find the friend they've been looking for:
Horton: You called my name and you set me free. One small voice in the universe...
JoJo: One true friend in the universe...
Both: Who believes in me.
In the finale, there's a quick reprise of this with Horton and Gertrude: