Taking the definition of refractive index $n = c/v$ into account, $n > 2$ means the velocity of light in the material $v$ is less than half than in vacuum $c,$ so there must be a strong interaction with the material.
One of the highest refractory index in liquids is $1.79~(\pu{20 °C})$ possessed by a solution of sulfur in methylene iodide $(\ce{CH2I2}).$ Liquids are less dense than solids and this explains why it's difficult to have a refractory index greater than 2. According to Anderson and Payne [1]:
Selenium monobromide, $\ce{Se2Br2}$, has a higher refractive index than that of any pure liquid hitherto recorded. Prepared by direct combination, the value for $\nu$ is $1.96±0.01$ rising to $2.02$ on exposure to the
atmosphere, owing to decomposition of the bromide, with separation and reabsorption of selenium.
From a more recent paper [2], selenium monobromide is confirmed as one of the liquid with the highest refraction index $(n > 2).$ However, diiodomethane is more easily accessible commercially.
A promising candidate solvent is the organic liquid diiodomethane $(\ce{CH2I2}),$ which is one of the liquids with the highest known refractive index values $(n = 1.74).$ While other high refractive index liquids exist (phenyldiiodoarsine $(\ce{C6H5AsI2})$ with $n = 1.85$ and selenium monobromide $(\ce{Se2Br2})$ with $n = 2.1),$ diiodomethane has the key advantage of being commercially available. In addition, diiodomethane is an excellent solvent, and many liquid formulations using salts dissolved in diiodomethane are reported to increase the refractive index and are even available commercially.
References
- Anderson, B. W.; Payne, C. J. Liquids of High Refractive Index. Nature 1934, 133 (3350), 66–67. DOI: 10.1038/133066b0.
- Laskar, J. M.; Kumar, P. S.; Herminghaus, S.; Daniels, K. E.; Schröter, M. High Refractive Index Immersion Liquid for Superresolution 3D Imaging Using Sapphire-Based Aplanatic Numerical Aperture Increasing Lens Optics. Appl. Opt., 2016, 55 (12), 3165–3169. DOI: 10.1364/AO.55.003165.