Can a stem cell donor claim any of the expenses related to the donation whether medical testing, travel, hotel & transportation?

There are no deductions at the federal level.

Some states allow a tax deduction or credit for blood or organ donations.

Please read these two documents for more information:

https://bethematch.org/support-the-cause/donate-bone-marrow/donor-safety-and-support/donor-leave/

https://bethematch.org/uploadedfiles/bethematchcontent/support_the_cause/donate_bone_marrow/donor_sa...