Tokyo Mew Mew: "Mew" is both a homophone of "mu" (a Greek letter and biological term, fitting with the Little Bit Beastly cast) and an onomatopoeia for a cat (the main character is a Catgirl).
Yakitate!! Japan refers to the main character's signature "Japan" baked goods. ("Pan" means "bread" in both Japanese and Spanish.)
This troper likes how the title was adapted into Portuguese, "Amassando Ja-păo", which not only keeps the pun, but does so in a single language.
Gintama; the word on its own means "silver soul" and revolves around the main character Gintoki, but is a near homophone for the word "kintama" which means "testicles".
The title of Urusei Yatsura literally translates to "People from the Planet Uru", although the word "Urusei" (which comes from "urusai", meaning annoying) is also a Japanese colloquialism to tell someone to "shut up" and the title can be interpreted as "Hey guys, shut up!" Animeigo attempted to translate the pun by changing the title to Those Obnoxious Aliens for the short-lived English dub of the anime. Of course, the main character is Moroboshi Ataru, whose name literally means "hit by a falling star", so the title is only the beginning of the Hurricane of Puns.
The episode titles in Ichigo Mashimaro, at least in English, vary between rhymes, alliteration, and this. For examples of this: "Violent Night", "The Hat's Meow", "Attack of the Killer [ZZZs]", "Into Hot Water", "Sick Jokes", "The Matsuri" (a borderline example: a matsuri is a festival as well as the name of a main character), "Schooled", and "Heart Attacks". Parts of the "An Amusing Stew (Using Miu)" episodes have titles as well: "Lack of Acute Judgment", "Thumb War", "Thrown By the Goat", "Phony", "What Possesses Her", and "Sketchy".
The Street Fighter video games had a TV anime series titled Street Fighter II V (that's a roman numeral "two" and the letter "vee"). The title doesn't seem to mean much by itself at first, but "two vee" is pronounced almost similarly to "tee vee", as in a Street Fighter TV series. The "V" also stands for "Victory" and since "two" can be a homophone for "to", the title can also be read as Street Fighter To Victory.
Dub episodes are often a pun on the featured Pokémon, such as "To Master the Onix-pected" and "Turning Over a Nuzleaf".
Pokémon 3: Spell of the Unown refers to two things: 1) the fact that Unown are Pokémon based on the alphabet, and 2) their powers.
Other episodes still have punny titles without Pokémon names, like "Gotta Catch Ya Later!" (a pun on the franchise's early Catch Phrase), "Hoenn Alone" (using the name of the home region of the Advanced Generation arc - this is the first episode of said arc, even), or "Home is Where the Start Is" (one of the arc-transition episodes when Ash returns to Pallet Town, this one bridges the gap between AG and the Diamond and Pearl).
Done away with as of Best Wishes, however, which goes back to the style of the early episodes.
The American episodes sometimes go to "gems" such as "Doin' What Comes Natu-rally" and "Smells Like Team Spirit". Japan sometimes fall to this ("Do Coil Dream of Electric Mice!?")
Haré+Guu: its original title, "Janguru wa Itsumo Hare Nochi Guu", is usually translated as "The Jungle was always nice, then came Guu" (or "Haré always lived in the jungle, then came Guu" - the title uses the name of both main characters to make the pun). However, the last three words are regularly used in Japanese weather forecast, and can be interpreted as something like "clear with a chance of showers". Thus, the title's underlying meaning would be "The jungle is always clear with a chance of showers."
A.I. Love You: "ai" is the Japanese word for "to love," it is pronounced like the letter I, and the second letter being "I" just completes the phrase "I love you."
A rather ironic example is Girls und Panzer. The Japanese pronunciation of "panzer" is "panzu", while the word "panties" is pronounced as "pantsu", as Lampshaded in some of the show's trailers. The irony comes in the fact that this show has no panty shots at all (despite the presence of other forms of Fanservice).