  The ARCADE2 and LWA1 experiments have claimed an excess over the Cosmic
Microwave Background (CMB) at low radio frequencies. If the cosmological
high-redshift contribution to this radio background is between 0.1% and 22% of
the CMB at 1.42 GHz, it could explain the tentative EDGES Low-Band detection of
the anomalously deep absorption in the 21-cm signal of neutral hydrogen. We use
the upper limit on the 21-cm signal from the Epoch of Reionization ($z=9.1$)
based on 141 hours of observations with LOFAR to evaluate the contribution of
the high redshift Universe to the detected radio background. Marginalizing over
astrophysical properties of star-forming halos, we find (at 95% C.L.) that the
cosmological radio background can be at most 9.6% of the CMB at 1.42 GHz. This
limit rules out strong contribution of the high-redshift Universe to the
ARCADE2 and LWA1 measurements. Even though LOFAR places limit on the extra
radio background, excess of $0.1-9.6$% over the CMB (at 1.42 GHz) is still
allowed and could explain the EDGES Low-Band detection. We also constrain the
thermal and ionization state of the gas at $z = 9.1$ and put limits on the
properties of the first star-forming objects. We find that, in agreement with
the limits from EDGES High-Band data, LOFAR data constrain scenarios with
inefficient X-ray sources and cases where the Universe was ionized by stars in
massive halos only.
