Beethoven's 4th Piano Concerto - none better than Gould w/ Bernstein (LP)
Beethoven's 5th Piano Concerto - lots to choose from, tonight's choice is Michealangeli w/ Giulini (CD)
Mozart Piano Concerto No. 21 - Anda w/ Anda (LP)
Bartók's Concerto for Piano and Orchestra No. 2 - Anda w/ Fricsay (LP)
It's not a piano concerto per se, but I just love this piece called "Totentanz", by Franz Listz. I have 4-5 recordings, none too thrilling sound-wise. The Zimerman/Ozawa/Boston is good, as well as the Cohen/Neschling/OSESP.
Many people (including me) consider the Fleisher/Szell performances of the five Beethoven and two Brahms (1st in particular) piano concerti to be outstanding. You can get them in this box set
For the Grieg and Schumann Concerti, Fleisher/Szell is my first choice. Sound quality is much better in the Sony Great Performances (2004 remaster) CD. There is an older Great performances CD with a yellow newsprint cover that has
I'm recommending several Fleisher/Szell recordings because their collaboration in the 50s and 60s produced wonderful performances.
For Rach 2 and the Rhapsody on a theme of Pananini, I like Rubinstein/Reiner, Wang/Abbado and Solti/Solti and Boult.
The Mozart Piano Concerti from 14 through 27 are all great music. No. 9 too. I'd start with No. 20 and Rudolf Serkin/Szell in the Masterworks Expanded Edition with no. 19 would be my first choice. There are lots of fine performances for 9 and 14-27.
These recommendations are based on quality of performance. Their sound quality is adequate for me to enjoy the performances without giving a thought to sound quality.
Lots of classics of course, Tchaikovsky definitely deserving a mention too. For a slightly leftfield suggestion, I also really love the Scriabin Concerto as played by Demidenko with the BBC Symphony Orchestra/Lazarev:
I have many favorite piano concertos. The Piano Concerto No. 1 Tchaikovsky, Rachmaninov 2, the Schumann or Chopin 1. And piano concertos of the twentieth century, I'll take the 3 of Prokofiev.
However, the concerts I check again and again are the five Beethoven piano concertos. They, more than any other artistic work, I find the evolution break and shift to the contemporary world.