Ideas? What pain medication is least likely to cause constipation?

Tough question. As different people respond differently to each medication (assuming you are asking specifically about opioids). Non-opioids are least likely to cause it. The lower the dose opioid, the less likely the constipation. There are some medications, such as a peripheral acting Naltrexone injection that can relieve the constipation. However, all of the opioid class are likely to cause constipation.