Alternative to Cooper pairing in superconductivity BCS theory, in which electrons form Cooper pairs, has been successful in explaining conventional superconductivity. Physicists have adopted similar pairing mechanism in an attempt to explain high-temperature superconductivity (HTS).
Is there any HTS theory that is not based on pairing? If not, what experiments preclude the existence of such a theory?
 A: To my knowledge (if it has any value), there is no such modern theory without pairing. 
So the question is why, right ?
There is no proposal trying to explain high-temperature superconductors (HTS) without paired electrons for the simple reason that there are (at least supposed, let me know if you disagree) evidences for pairing ! If you know about the Aharonov-Bohm effect, you will not be shocked by two facts


*

*due to Aharonov-Bohm, there are some quantities which oscillates at the ratio $\Phi /2 \Phi_{0}$, with $\Phi_{0}=h/2e$ the superconducting flux quantum, $\Phi$ being the magnetic flux piercing some non-connected region of matter (metal or superconductors in our case of concern). So, for normal electrons some things actually oscillate with the characteristic value of the Planck constant $h$ over the charge of the electron $e$

*there is no known quantity oscillating at $\Phi /2 \Phi_{0}$ for superconductors, everything is oscillating at $\Phi / \Phi_{0}$, even for HTS, as for the Little-Parks effect for instance. In short, Little-Parks effect is the oscillation with the magnetic flux of the critical temperature of a superconducting plate with a hole in the middle. All these interfering experiments need the composite particle with charge $2e$ to be understood.
Unfortunately, I can not find some good references discussing this point. I'll try to include some latter. I believe there are some discussions in the book by A.J. Leggett Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems Oxford Graduate Texts.
edit Haha edit ! I've done the previous answer based on some discussions I had with colleagues. I was (stupidly) looking for Little-Parks effect + HTS on Google Scholar without result :-( Of course the good key words to find something are flux quantisation + HTS. The first result is already enough I believe 

Gough, C. E., Colclough, M. S., Forgan, E. M., Jordan, R. G., Keene, M., Muirhead, C. M., Rae, A. I. M., et al. Flux quantization in a high-Tc superconductor. Nature, 326(6116), 855–855 (1987).

Nevertheless, the Leggett's book (that I read when it was first published and forget then) should contains a lot of material in one of its last section. Leggett discusses there all what we know about cuprates. 
