If you are a volunteer enrolled under a research protocol (you will know if you are because you will have to read and sign an Informed Consent document to participate), you should not have to pay for anything (i.e. free!). All expenses should be covered by the research grant supporting the study.
If you are having a clinical exam, you will not be enrolled in a research study. You (or your insurance company) will have to pay for the exam. Be warned that since breast MRI is an evolving field and there are not the same standard practices as with mammograms, your insurance company may not cover this exam.
Between the costs of the magnet time, the technologists running the scan, the nurse or doctor inserting the I.V. and injecting the contrast agent, and the contrast agent itself, the entire exam can cost $1000-$1500.