Whew... Such a question could keep me around for the rest of the day... There are so many, many melodies I can count among my all-time favorites, that listing them all could be a long, time-consuming task.
Nevertheless, I could give a few answers (I promise to post the links as I'm able to search them all Links added!).
Short answer: almost anything by Tchaikovsky.
Slightly longer answer, in no particular order:
1) The first few measures of Rachmaninov's 2nd Symphony, Op. 27, 3rd movement.
2) The soaring love theme from Tchaikovsky's The Tempest, Op. 18 (this section starting at 16:27)
3) The horn solo from Tchaikovsky's 5th Symphony, Op. 64, 2nd movement. (Yeah, call it overplayed, but I'm just answering a question). Starts at 1:07.
4) The main themes from Mahler's 4th Symphony, 1st movement.
5) The opening theme from Wagner's Rienzi Overture (starts at 1:32).
6) Don Jose's Aria, from Bizet's Carmen, Act II.
7) Tchaikovsky's Rose Adagio, from The Sleeping Beauty, Op. 66, Act I (starting at 1:13).
8) John Williams's Flying Theme, from the score for ET.
9) The second theme from Tchaikovsky's Orchestral Suite N° 3, Op. 55, 1st movement (starting at 2:32).
10) The full 2nd movement from Beethoven's Pathetique Sonata, Op. 13.
11) Haendel's For Unto Us a Child Is Born chorus, from Messiah, Part I.
12) Rachmaninov's Vocalise.
13) Air on the G string, from Bach's Orchestral Suite N° 3. (Again, call it overplayed or whatever)...
14) Puccini's famous aria Nessun Dorma, from Turandot. (Likewise).
15) Hymn and Chorus from Tchaikovsky's The Maid of Orleans, Act I (starts at 11:25).
16) The closing melody from Tchaikovsky's Letter Scene, from Eugene Onegin, Act I.
17) The second theme from Tchaikovsky's Manfred Symphony, Op. 58, 2nd movement (starts at 22:58).
18) The main theme from Mendelssohn's Violin Concerto, Op. 64, 1st movement.
To be continued...