A blurb from Austral Piano World, an Australian Bechstein dealer's website, http://www.pianoworld.com.au/main-bechstein.html
And before him Franz Liszt...
the unsurpassed king of the keyboard, celebrated by the whole of Europe as a piano genius. He translated the unheard-of violin virtuosity of the legendary Paganini to the piano. The great magician gave decisive stimulus to the continued development of the Bechstein grand piano. Before Bechstein pianos were available, he had sometimes needed several grand pianos in one evening - just one was not enough for the requirements of his majestic playing. Carl Bechstein was the first piano maker to supply an instrument that understood his language and enabled his powerful temperament to find full freedom of expression. Bechstein became Liszt's lifelong instrument. His pupils also followed his example, particularly the great Hans von Bulow, the first Chief Conductor of the Berlin Philharmonic Orchestra.
A Liszt quote about Bechstein pianos (the one they use in their promotional literature): "I have now played your instruments for 28 years, and you have maintained your supremacy."
Historian, Jeffrey Dane, wrote an article called "Pianos of Beethoven and Other Famous Composers" that mentions Liszt's Bechstein. See http://www.frugalfun.com/pianohistory.html
"A legendary (and documented) characteristic of Liszt's piano-playing was a tone of distinctive and peculiarly effective carrying quality. Today at his house (now a museum) on Marienstrasse in Weimar, Germany, one finds two pianos in the music room, the keyboards nearly facing each other, with a piano stool between them: a huge seven and a half octave, Berlin-made Bechstein concert grand, (acquired by Liszt in 1869) which dominates the room, and, against a wall behind it, an upright piano made in Cologne by Rudolf Ibach & Sohn (now located in Schwelm), a gift to Liszt in 1885. At one point during the years he lived in Weimar, Liszt had his own personal "collection" of pianos: in addition to the two now on display, he had a concert Erard, Beethoven's Broadwood, and a piano once owned by Mozart. It may have been the enormous Bechstein which Liszt used when composing his last piano works (such as Nuages Gris), some of the harmonies of which belong to the dawn of 20th-century music."
I saw Liszt's Bechstein on display at the Bechstein Centrum in Berlin during a visit there a few years ago. No, they wouldn't let me play it. I've got a picture of it somewhere.
As others have mentioned, Liszt had many pianos over the years. Piano manufacturers were falling over each another to associate their pianos with his name. It wouldn't surprise me if Liszt was given a piano by every important piano maker of his day. However, Liszt played a Bechstein most his professional life, and as Rich mentioned, he was fond of Bösendorfers too.