  Aims. We present a spectroscopic analysis of the extremely luminous red star
VX Sgr based on high-resolution observations combined with AAVSO light curve
data. Given the puzzling characteristics of VX Sgr, we explore three scenarios
for its nature: a massive red supergiant (RSG) or hypergiant (RHG), a Thorne
Zytkow object (TZO), and an extreme Asymptotic Giant Branch (AGB) star.
  Methods. Sampling more than one whole cycle of photometric variability, we
derive stellar atmospheric parameters by using state-of-the-art PHOENIX
atmospheric models. We compare them to optical and near infrared spectral
types. We report on some key features due to neutral elemental atomic species
such as Li I, Ca I, and Rb I.
  Results. We provide new insights into its luminosity, evolutionary stage as
well as its pulsation period. Based on all the data, there are two strong
reasons to believe that VX Sgr is some sort of extreme AGB star. Firstly, its
Mira-like behaviour during active phases. VX Sgr shows light variations with
amplitudes much larger than any known RSG and clearly larger than all RHGs. In
addition, it displays Balmer line emission and, as shown here for the first
time, line doubling of its metallic spectrum at maximum light, both
characteristics typical of Miras. Secondly, unlike any known RSG or RHG, VX Sgr
displays strong Rb I lines. In addition to the photospheric lines that are
sometimes seen, it always shows circumstellar components whose expansion
velocity is compatible with that of the OH masers in the envelope,
demonstrating a continuous enrichment of the outer atmosphere with s-process
elements, a behaviour that can only be explained by third dredge up during the
thermal pulse phase.
