The standard genetic code is robust to mutations and base-pairing errors during transcription and translation. Point mutations are most likely to be synonymous or preserve the chemical properties of the original amino acid. Saturation mutagenesis experiments suggest that in some cases the best performing mutant requires a replacement of more than a single nucleotide within a codon. These replacements are essentially inaccessible to common error-based laboratory engineering techniques that alter single nucleotide per mutation event, due to the extreme rarity of adjacent mutations. In this theoretical study, we suggest a radical reordering of the genetic code that maximizes the mutagenic potential of single nucleotide replacements. We explore several possible genetic codes that allow a greater degree of accessibility to the mutational landscape and may result in a hyper-evolvable organism serving as an ideal platform for directed evolution experiments. We then conclude by evaluating potential applications for recoded organisms within the synthetic biology field.