(Unfortunately, this doesn't directly answer the "showers per day" question, but does address the more general issue of over-washing.)
Repetitive washing can cause Transepidermal Water Loss (TEWL):
All skin function parameters already showed a marked change after a single wash (e.g., the median of TEWL values increased by more than 0.5 g/m2 h). Repetitive washing for 1 week led to a further deterioration of TEWL and corneometry values (e.g., TEWL increased for about 2.9 g/m2 h after repetitive washing with sodium lauryl sulfate).
In another experiment (intended to test hand-creams) they reproduced this effect to act as a control:
Twenty-five subjects washed hands and forearms with a neutral soap four times per day, for 2 minutes each time, for a total of two weeks.
They measured skin roughness and hydration:
Washing hands lead to a gradual increase of skin roughness from 100 (baseline) to a maximum of 108.5 after 9 days. [...] Skin hydration was gradually decreased after washing hands from 79 (baseline) to 65.5 after 14 days.
Part of their conclusion was:
Repetitive and frequent hand washing increases skin dryness and roughness.