  Cosmological constraints are usually derived under the assumption of a $6$
parameters $\Lambda$-CDM theoretical framework or simple one-parameter
extensions. In this paper we present, for the first time, cosmological
constraints in a significantly extended scenario, varying up to $12$
cosmological parameters simultaneously, including the sum of neutrino masses,
the neutrino effective number, the dark energy equation of state, the
gravitational waves background and the running of the spectral index of
primordial perturbations. Using the latest Planck 2015 data release (with
polarization) we found no significant indication for extensions to the standard
$\Lambda$-CDM scenario, with the notable exception of the angular power
spectrum lensing amplitude, $A_{\rm lens}$ that is larger than the expected
value at more than two standard deviations even when combining the Planck data
with BAO and supernovae type Ia external datasets. In our extended cosmological
framework, we find that a combined Planck+BAO analysis constrains the value of
the r.m.s. density fluctuation parameter to $\sigma_8=0.781_{-0.063}^{+0.065}$
at $95 \%$ c.l., helping to relieve the possible tensions with the CFHTlenS
cosmic shear survey. We also find a lower value for the reionization optical
depth $\tau=0.058_{-0.043}^{+0.040}$ at $95$ \% c.l. respect to the one derived
under the assumption of $\Lambda$-CDM. The scalar spectral index $n_S$ is now
compatible with a Harrison-Zeldovich spectrum to within $2.5$ standard
deviations. Combining the Planck dataset with the HST prior on the Hubble
constant provides a value for the equation of state $w < -1$ at more than two
standard deviations while the neutrino effective number is fully compatible
with the expectations of the standard three neutrino framework.
