We know that there is uncertainty principle, so question: can we ever measure wavefunction of particles? I do not think this is possible, but I am not sure. I guess that everything is probabilistic. (that's why, I believe, we have sigma level when we say we discovered some particles...)
If you are able to produce multiple copies of the same pure quantum state, then it is possible to reconstruct the wavefunction. In that case, you need a relatively precise experiment, as just measuring the position and building a histogram will only give you the mod-squared of the wavefunction. To get some information of the phase, you might try measuring the momentum distribution, which tells you about the mod-squared of the wavefunction's Fourier transform, but that comes up a bit short; to get the full quantum state you need to perform a state tomography protocol.
If you only have a single particle, getting the quantum state is impossible, because it contains more information than you can get in any one measurement and that one measurement will destroy the state. Even for a qubit, where only two states (and coherent superpositions of them) are allowed, any tomography protocol needs at least two measurements, with a large number of repetitions determined by the required precision.
Now, as to whether you can "measure" the "wavefunction", it is still a complicated open problem in quantum foundations. Operationally, in a tomographical protocol what you're doing is essentially diagnosing your preparation procedure, and there's no one right now that even pretends to know for sure whether the quantum state is in fact a physical quantity, or whether it is an epistemic thing related to our statistical knowledge of quantum systems.
According to this paper, an experiment was performed that measured the single electron's physical wave function by causing it to interfere with itself. The interference pattern matched the predictions of the Schrodinger equation. So, apparently this was a direct measurement of an electron's wave. Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States A. S. Stodolna, A. Rouzée, F. Lépine, S. Cohen, F. Robicheaux, A. Gijsbertsen, J. H. Jungmann, C. Bordas, and M. J. J. Vrakking Phys. Rev. Lett. 110, 213001 (2013) Published May 20, 2013
I also believe that Bose Einstein Condensates are examples of macroscopic quantum wave functions. Since it occurs when the wave functions of individual atoms "overlap" thus forming a single matter wave which evolves according to Schrodinger's equation. However, I believe this begs the question as to what is actually occurring. If the wave function is physical real but only collapses when there is some "decoherence", then physics needs to explain this mechanism of wave function collapse. Furthermore, tests of BEC's show that the matter waves behave both classically and quantum mechanically based on whether all of the atoms in the condensates are accounted for by the experiment.(http://www.lkb.ens.fr/IMG/pdf/Toulouse-Leggett_April2012-2.pdf)
You can measure the amplitude squared of the wave function if you have many copies of the system. You can then make a histogram of the recorded observations of the systems. This histogram will tend to the amplitude squared of the wave function as the number of copies tends to infinity.