We have started an ambitious program to determine if the full diversity of extinction laws is real or if some of it is due to calibration or methodological issues. Here we start by analysing the information on near-infrared (NIR) extinction in a Two Micron All-Sky Survey (2MASS) stellar sample with good quality photometry and very red colours. We calculate the extinction at 1 μm, A1, and the power-law exponent, α (Aλ = A1λ-α), for the 2MASS stars located in the extinction trajectory in the H - K versus J - H plane expected for red giants with A1 > 5 mag. We test the validity of the assumption about the nature of those stars, whether a single or multiple values of α are needed, and the spatial variations of the results. Most (∼83 per cent) of those stars can indeed be explained by high-extinction red giants and the rest are composed of extinguished asymptotic giant branch (AGB) stars (mostly O-rich), blended sources, and smaller numbers of other objects, a contaminant fraction that can be reduced with the help of Gaia Data Release 2 (DR2) data. Galactic red giants experience a NIR extinction with α ∼ 2.27 and an uncertainty of a few hundredths of a magnitude. There is no significant spread in α even though our sample is widely distributed and has a broad range of extinctions. Differences with previous results are ascribed to the treatment of non-linear photometric effects and/or the contaminant correction. Future research should concentrate in finding the correct functional form for the NIR extinction law. In the appendix, we detail the treatment of non-linear photometric effects in the 2MASS bands.