Is the total information available in the universe compatible with the entropy we would find on the surface of an hypothetical black hole whose radius is the visible universe one ($\sim14$ Gpc)?
We know that the entropy of a BH is proportional to its surface. I would love to discuss the possibility that we are actually living inside a BH, just in a toy model... Of course keeping into account the actual BBR at 2,7 K. Is a totally insane direction?
I won't comment on the "actually living inside a BH" idea, but here are some numeric comparisons, using entropy to quantify "information," and interpreting "radius" as "Schwarzschild radius."
According to [1], the entropy of the observable universe (excluding black holes) is dominated by the cosmic microwave background photons. Based on this, and using $$ 1\text{ parsec}\sim 3\times 10^{16}\text{ meters}, \tag{1} $$ we can very roughly estimate the entropy of the observable universe to be $$ S_\text{CMB}\sim \left(\frac{14\times 10^{9}\text{ parsecs}}{1\text{ millimeter}}\right)^3 \sim 10^{89}. \tag{2} $$ Inside the parentheses, the numerator is the radius of the observable universe, and the denominator is the wavelength of a typical CMB photon [2]. This is only a very crude estimate, of course.
For comparison, the Bekenstein-Hawking entropy associated with a black hole [3] is $$ S_\text{BH} = \frac{c^3}{\hbar G}\,\frac{A}{4} = \frac{A}{4L^2} \tag{3} $$ where $A$ is the area of the horizon with Schwarzschild radius $R$, and $$ L=\sqrt{\hbar G/c^3}\sim 1.6\times 10^{-35}\text{ meter} $$ is the Planck length. According to equation (3), the entropy of a solar-mass black hole ($R\approx 3$ km) is $\sim 10^{77}$, and the entropy of a black hole with Schwarzschild radius $$ R\sim 14\times 10^9\text{ parsecs} \sim 4\times 10^{26}\text{ meters} $$ would be $$ S_\text{BH} \sim 10^{123}. $$ According to these estimates, the entropy of the observable universe, $\sim 10^{89}$, is much less than the entropy of a universe-sized black hole, $\sim 10^{123}$. I am practicing the art of understatement.
Reference:
[1] Section 4.5 in Harlow (2014), "Jerusalem Lectures on Black Holes and Quantum Information," http://arxiv.org/abs/1409.1231
[2] "CMB Spectrum," https://asd.gsfc.nasa.gov/archive/arcade/cmb_spectrum.html
[3] Sections I and II in Bousso (2002), "The holographic principle", https://arxiv.org/abs/hep-th/0203101
