Final answer:
Polycystic ovary disease (PCOS) is the most likely condition to contribute to the infertility of the obese woman with hirsutism, irregular menstrual cycles, and a history of gonorrhea infection.
Step-by-step explanation:
The most likely condition to contribute to the infertility of the obese woman with hirsutism, irregular menstrual cycles, and a history of gonorrhea infection is Polycystic ovary disease (PCOS). PCOS is a hormonal disorder that can affect a woman's ability to produce viable eggs and ovulate regularly, leading to infertility. The excess production of androgens and insulin resistance associated with PCOS can disrupt the menstrual cycle and interfere with normal ovulation.