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Obviously, Han understands Chewbacca (almost) perfectly. In the prequel trilogy, Yoda seems to understand him too. In at least one scene in the original trilogy, C-3PO also understands Chewie, which is logical, since - as a protocol droid - translation is one of 3PO's primary functions.
THREEPIO: Captain Solo, this time you have gone too far.
(Chewie growls)
THREEPIO: No, I will not be quiet, Chewbacca. Why doesn't anyone listen to me?
From the script for A New Hope:
Ben is standing next to Chewbacca, an eight-foot-tall savage-looking creature [...] He is a two-hundred-year-old Wookiee and a sight to behold.
Ben speaks to the Wookiee, pointing to Luke several times during his conversation and the huge creature suddenly lets out a horrifying laugh. Luke is more than a little bit disconcerted and pretends not to hear the conversation between Ben and the giant Wookiee.
From the novel:
All but old Ben - Ben, who was talking to the Wookiee in its own language, quarreling and hooting softly like a native.
As shown in this answer from @phantom42 on another site, Rey has at least some understanding of Shyriiwook. Quoting from the novelisation of The Force Awakens:
She tried to find something to say to that, something worthy of the sentiment and the risk they had undertaken. She failed miserably. Chewie, however, had something of his own to add. Whatever the Wookiee had uttered caused tears to well up in her eyes. Having never found himself in such a position before, Finn was unsure how to respond. Knowing well her inner toughness, he wondered what Chewbacca had said that could have inspired such a reaction.
“What’d he say?”
She sniffed and wiped at her face. “That it was your idea.”
"...it's rough on my throat, I lost my voice for a day after bugging Zeb by serenading him in Shyriiwook."
-- Sabine Wren, Star Wars Rebels: Sabine, My Rebel Sketchbook
When he told them the problem, the Wookiees nodded and barked at each other so fast that it was difficult even for him to follow the conversation. Then they turned and offered him one of their escape pods. Yoda accepted at once. The only problem left was how to get past the clone troops to the pod.
“According to her diplomatic profile, Princess Leia speaks both Huttese and Shyriiwook fluently. Regretfully, and as Your Majesty is keenly aware, the princess is merely proficient in Shyriiwook. This is, naturally, a failure of mine and not the princess, but I worry that such an error might cause her embarrassment. Oh, it is too, too humiliating.”
If we also include the Star Wars Holiday Special, then we can also add
Chewie's wife Malla, his son Lumpy and his father Itchy as well as human trader Saun Dann, all of whom demonstrate an ability to understand Chewbacca.
Note: I don't like getting rep from self-answers, so this is a Community Wiki answer.
One of the other main characters from episodes V and VI, Lando Calrissian, speaks Shyriiwook well enough to bicker with C-3PO over a translation:
Chewie said something. Nobody spoke for a moment. Leia said, “Well, is somebody going to tell me what he said?”
Threepio got it out first: “Chewbacca says the woman made him very nervous.”
“He didn’t say ‘very,’” Lando said. “Just plain ‘nervous.’ ”
“Excuse me,” Threepio said. “I inferred the modifier from his tone. Wookieespeak allows for such shadings.”
“You saying my Wookieespeak is bad?” Lando said.
- Star Wars: Shadows of the Empire
Throughout this book, Lando shows his proficiency in understanding Shyriiwook on a regular basis.
