We present a novel class of Multi-Level Markov Chain Monte Carlo (ML-MCMC) algorithms and apply them in the context of large-scale Bayesian inverse problems (BIPs), where the likelihood function involves a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard Multi-level Monte Carlo argument to obtain a better cost-tolerance complexity than a single-level MCMC algorithm. We present several approaches to generate these highly coupled chains by sampling, e.g., from a maximal coupling of the proposals for each marginal Markov chain. By doing this, we are allowed to create novel ML-MCMC methods for which, contrary to previously used models, the proposals at each iteration can depend on the current state of this chain, while at the same time, creating chains that are highly correlated. The presented methods are tested on an array of both academic and large-scale BIPs, where a clear computational advantage can be observed with respect to their single-level counterpart.