The theme "Nunnally" from Code Geass uses the same melody, same lead instrument, SAME KEY, and a similar chord progression to the main theme from an obscure Ennio Morricone-scored film, Malena. Hear foryourself.
Similarly, Pedro's music is the guitar riff from "House Of The Rising Sun."
The second episode uses a parody of Metallica's "Sad but True."
Nandemo Q was a children's television show which aired on NHK between 1995 and 2004 that featured educational Suspiciously Similar Songs of popular western songs that often featured Pun Based Titles such as "Hotaru California".
Full Metal Panic!: "Tokkou Yarou", a background theme used in that series is incredibly similar to the theme from The A-Team. Lampshaded with a Theme Tune Cameo as a cellphone ring tone and someone says it's sounds like an old American TV show (which indeed was called Tokkou Yarou A-Team when broadcast in Japan).
Another example ("Barefoot in the Park") is suspiciously similar to "The Girl from Ipanema" used when the characters are having a coffee break.
Shinji's theme, "I. Shinji" is basically a minor-key version of "God Save the Queen" mashed-up with "The Godfather Waltz".
More recently, on the soundtrack for Evangelion 2.0: You Can (Not) Advance, the track "2EM05_KK_B09" sounds suspiciously similar to "The Third Man Theme" by Anton Karas.
Another example of this, also from the soundtrack of "Evangelion 2.0: You Can (Not) Advance", in which the track "L'Agresseur (2EM03)"  sounds pretty much the same like "Dennis steals the Embryo"  from the soundtrack to Jurassic Park.
"Misato's Theme" also shadows parts of "Why Do Fools Fall In Love" which has been covered by artists like Diana Ross.
"Komm, Susser Todd" or that song that plays when everyone gets turned into Tang - sounds a lot like "Hey, Jude."
The slide guitar solo at the end of the song owes a bit to the ending solo of Pink Floyd's High Hopes.
A closer similarity is within the original version of the series itself, and intentional: Tsubasa wo Daite, Michel's Villain Song, and Ashita ga Mienakute, Michal's Image Song. Their names already matched, as did the names of their birds, and they couldn't be active at the same time... but when their songs matched, all bets about coincidence were off.
The Training Montage in the first episode of GunBuster uses a musical cue essentially ripped off from Vangelis' theme from Chariots of Fire. So much so, in fact, that Studio Gainax ordered the US distributors to replace the music in the North American re-release.
Public Domain Example: The last episode uses a conspicuously-synthesised piece of music that sounds suspiciously similar to the Hallelujah chorus. Considering that the original song would later turn up in Neon Genesis Evangelion, this example's rather perplexing.
In an earlier example of ripping off Vangelis, the music from the first scene in the original Bubblegum Crisis OVAs is suspiciously similar to the opening theme from Blade Runner.
The intros to "Victory" and "Akuma to Tenshi no Kiss" are derived from different parts of the intro to "Nowhere Fast", the opening theme from Streets of Fire.
The guitar riff in "Say Yes" from OVA 7 resembles U2's "Where the Streets Have No Name".
"Konya wa Hurricane" is very clearly an attempt to create a song in the style of Jim Steinman, particularly those from Streets Of Fire.
Near the end of the Ah! My Goddess OVA, a Suspiciously Similar Song to "L'amour est bleu" shows up in the scene where Keiichi works several part-time jobs to buy Belldandy a ring.
On the soundtrack for Mobile Suit Gundam 0083: Stardust Memory, Mitsuo Hagita directly copies, note for note, extensive parts of two pieces from James Horner's Brainstorm score: "The Battle of Solomon Sea" was Brainstorm's "Race For Time", and "Burn Down" was "Lillian's Heart Attack".
"The Winner" (especially the English version sung by Ritchie Kotzen) from the same show, sounds somewhat like "Danger Zone" from Top Gun.
A Suspiciously Similar Song to Pachelbel's Canon plays during the graduation ceremony in K-ON!! episode 24, which is peculiar because, again, it's in the public domain.
When the music store clerk tells the history of Sawako's guitar, images of Eric Clapton, Angus Young, Tony Iommi and Pete Townshend are shown, along with Suspiciously Similar Songs to Sunshine of Your Love, Thunderstruck, Paranoid, and Won't Get Fooled Again respectively.
"A New Foe Rears His Head" is just as blatantly a ripoff of the Terminator Salvation opening theme. In a way, this one is almost forgivable; even the original show had a song that sounded suspiciously similar to the Terminator theme, which was used in a scene that was lifted straight from the movie, in the middle of a story arc that was taken straight from the movie (the Trunks and Android sagas, of course.) Can you really blame the composer for thinking originality wasn't necessary?
One of the pieces of music from Read or Die sounds like the stock music piece "Valse Aux Champs-Elysees", which was used on Ren and Stimpy.
That's minimal, Read or Die's main theme rips off from every musical aspect of James Bond's main theme... without actually ripping a note from it, and made a much more cohesive, surprising and captivating version, like Picasso (allegedly) said "Good artists borrow, great artists steal".
The first ending theme in the English dub version of Jungle wa Itsumo Hare nochi Guu sounds suspiciously like the soul classic "T.S.O.P. (The Sound of Philadelphia)".
The opening song of the Yu-Gi-Oh! GX English dub sounds an awful lot like Blink 182's "Dammit" at times.
One of the songs from Axis Powers Hetalia , "Carrots and Sticks", is sung by Belarus and Ukraine. If you pay attention to the background music, it sounds like a pitched-down, remade version of t.A.T.u.'s All The Things She Said.
To say nothing of the music cue for the opening titles, which strongly evokes "Also Sprach Zarathustra" (AKA the 2001: A Space Odyssey theme).
The Monster anime has "Over The Rainbow" played multiple times; it's even explicitly referred to as such. Except, it's instrumental and only vaguely sounds like it. The song isn't in the public domain. I don't know if the original Japanese version had this problem or not.
The Kodomo No Jikan ending "Hana Maru Sensation" sounds almost exactly like the chorus from The Offspring's One Fine Day, which in turn sounds like the chorus of the Kiteretsu Daihykka theme song "Suimin Bosukou".
In Zoids: Chaotic Century, the theme that plays when the Ultrasaurus awakens sounds an awful lot like the Jurassic Parktheme. They sound almost identical when both themes climax. Also consider how similar the scenes with the Ultrasaurus first moving and Dr. Grant seeing the living Brachiosaur are, especially since the aforementioned themes start playing in both instances.
Skip Beat!! is an interesting example in that it's a plot point when some random band comes out of nowhere and plagiarizes Fuwa Sho's released work and make it big. No one in the industry comments, and Sho ends up getting only more and more pissed. Then Kyoukoenters the picture.
In the episode of Doki Doki Pretty Cure introducing the Love Heart Arrow, when the girls introduce themeselves, rock music that sounds like this U.S. Acres song plays as background music.