The MUSE Hubble Ultra Deep Field Survey X. Ly$\alpha$ Equivalent Widths
  at $2.9 < z < 6.6$
  We present rest-frame Ly$\alpha$ equivalent widths (EW) of 417 Ly$\alpha$
emitters (LAEs) detected with Multi Unit Spectroscopic Explorer (MUSE) on the
Very Large Telescope (VLT) at $2.9 < z < 6.6$ in the Hubble Ultra Deep Field.
Based on the deep MUSE spectroscopy and ancillary Hubble Space Telescope (HST)
photometry data, we carefully measured EW values taking into account extended
Ly$\alpha$ emission and UV continuum slopes ($\beta$). Our LAEs reach
unprecedented depths, both in Ly$\alpha$ luminosities and UV absolute
magnitudes, from log($L_{\rm Ly\alpha}$/erg s$^{-1}$) $\sim$41.0 to 43.0 and
from Muv $\sim$ -16 to -21 (0.01-1.0 $L^{*}_{\rm z=3}$). The EW values span the
range of $\sim$ 5 to 240 \AA\ or larger, and their distribution can be well
fitted by an exponential law $N = N_{\rm 0}$ exp($-$EW/$w_{\rm 0}$). Owing to
the high dynamic range in Muv, we find that the scale factor, $w_{\rm 0}$,
depends on Muv in the sense that including fainter Muv objects increases
$w_{\rm 0}$, i.e., the Ando effect. The results indicate that selection
functions affect the EW scale factor. Taking these effects into account, we
find that our $w_{\rm 0}$ values are consistent with those in the literature
within $1\sigma$ uncertainties at $2.9 < z < 6.6$ at a given threshold of Muv
and $L_{\rm Ly\alpha}$. Interestingly, we find 12 objects with EW $>200$ \AA\
above $1\sigma$ uncertainties. Two of these 12 LAEs show signatures of merger
or AGN activity: the weak CIV $\lambda 1549$ emission line. For the remaining
10 very large EW LAEs, we find that the EW values can be reproduced by young
stellar ages ($< 100$ Myr) and low metallicities ($\lesssim 0.02$ $Z_{\rm
\odot}$). Otherwise, at least part of the Ly$\alpha$ emission in these LAEs
needs to arise from anisotropic radiative transfer effects, fluorescence by
hidden AGN or quasi-stellar object activity, or gravitational cooling.
