Accumulating evidence indicates that some protein-coding genes have originated de novo from previously non-coding genomic sequences. However, the processes underlying de novo gene birth are still enigmatic. In particular, the appearance of a new functional protein seems highly improbable unless there is already a pool of neutrally evolving peptides that are translated at significant levels and that can at some point acquire new functions. Here, we use deep ribosome-profiling sequencing data, together with proteomics and single nucleotide polymorphism information, to search for these peptides. We find hundreds of open reading frames that are translated and that show no evolutionary conservation or selective constraints. These data suggest that the translation of these neutrally evolving peptides may be facilitated by the chance occurrence of open reading frames with a favourable codon composition. We conclude that the pervasive translation of the transcriptome provides plenty of material for the evolution of new functional proteins.