A very plain definition of chords would be, three or more notes played together and which sound musical right. So could there be chords that have so many notes in them (>10) that they cannot be played in a piano or any instrument.
This question is trivially answered when more than one player may be involved. Even with one piano player, there is always the pedal for letting notes sound on without tieing up fingers. With organs, there are dedicated small wedges you may put between keys to hold a key down without human intervention: that way even a single player can accumulate complex chords on the keyboard (and registrations/manual couplings make for additional multiplicity of notes in some sense of the word).
A very plain definition of chords would be, three or more notes played together and which sound musical right.
I wish. Take out the part about sounding right and we've got something to talk about. A chord is just 3 or more notes played together regardless of what they sound like, however, there is fine print to this.
Suppose I play a 440Hz A. I can now play a 660Hz E and a 700Hz F to complete a chord (and it won't sound too good). But I can also play along with the 440Hz A a 442Hz A and a 438Hz A. Is this still a chord? The notes are essentially the same because of the approximate mapping from frequencies to notes, but the frequncies are not. Turns out it's not considered a chord because of the musical context when talking about chords, especially chord progressions which looks at their function and not their acoustic properties. Just 3 (or more) different "pitched sounds" are not enough.
Could there be chords that have so many notes in them (>10) that they cannot be played in a piano or any instrument.
From the above discussion, it's already clear that the definition and usage are theoretical, so the means of production are irrelevant. A player with polydactylism could play 11 or 12 notes, or 20 notes in pieces for 4 hands piano. There are even "special chords" (predefined chords with names like French 6th and Mystic chord) with more than 10 notes like the Northern lights chord, which, by the way, sounds terrible.
So not only there could be chords with more than 10 notes in them, there are and the number 10 turns out to be completely arbitrary as you can have 1000 people play different notes at the same time.