Can we look further than the CMB? As I understood it, the map of the cosmic microwave background (CMB) corresponds to the time when photons decoupled from matter, making the universe transparent. In this sense, we might expect the CMB to be a limit to how far we can see.
Nevertheless, is it possible to say that we can look further than that? For example, if we look at the CMB fluctuations, or at the time-evolution of the CMB spectrum, can we infer/look beyond the time at which photons decoupled? 
EDIT: as mentioned in a comment, what I was suggesting is that observing something other than electromagnetic radiation would allow to go further. The standard model does not give many options, and gravitational waves seems to be the best one.
 A: Yes. We can either look for the cosmic neutrino background or for gravitational waves associated with the big bang. Both of these probe the conditions considerably earlier than the the 380,000 years after the big-bang probed by the CMB.
Neither of these are fanciful ideas - there are good theoretical reasons to expect both to exist, and in both cases there are plans in progress to develop instumentation to detect them.


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*The cosmic neutrino background is formed in a similar way to the cosmic microwave background. It occurs when the universe is $\sim 1$ second old, the temperature has fallen to $\sim 3\times 10^{10}$K and the matter becomes transparent to neutrinos. The decoupled neutrinos are initially ultra-relativistic, but their de Broglie wavelengths are stretched by the expansion of the universe, such that they are expected to have energies of order  0.1 meV now. The low energy neutrino background might be detectable by the capture of electron neutrinos by tritium nuclei, transforming it to helium via beta decay. A careful analysis of the beta electron energy spectrum (e.g. Faessler 2016). Such instruments are being built - e.g. the KATRIN neutrino experiment might be capable of detecting this signature, but really needs a bigger tritium source and better spectral resolution. A better bet will be the PTOLEMY experiment (Betti et al. 2019), which uses a similar principle and is currently in a research and development phase.

*The cosmological gravitational wave background is discussed by Caprini & Figueroa (2018). Gravitational waves are decoupled from matter as soon as they are produced, at any epoch, right back to the inflationary era. They may be produced by various processes, including inflation, particle creation, primordial black holes, modified gravity theories and many more. There is expected to be a continuous spectrum of gravitational wave frequencies, possibly featuring multiple power laws and/or "knees" in the distribution. Their detectability is possible at a very wide range of frequencies. At present, only the range 10-1000 Hz is sampled by the LIGO and VIRGO experiments, but space-based instruments like LISA and ground-based pulsar timing arrays can extend this sensitivity to much lower freqencies.
