Any suggestions for puffy nipples and chest fat? (photos)

It's possible you have a small amount of gynecomastia.  But it is impossible to tell based on the photograph.  If this is something that bothers you greatly you should seek out a plastic surgeon for an evaluation.  More info below on gynecomastia.

"Puffy Nipple Areola Complex" is among the most common forms of gynecomastia. This gland and/or adipose tissue accumulation can be located under the areola or can be slightly extended outside the areola, causing the areola to appear dome shaped.This form of #Gynecomastia can be found in all age groups but is more common in young adults.As each case is different, it is vital that you consult with a board certified plastic surgeon who is experienced in male breast surgery. The surgeon should determine if you are a suitable candidate for male breast reduction by liposuction or glandular excision. It is also crucial to have a plastic surgeon who is well-versed in the gynecomastia condition can determine if there is firm breast tissue beneath the areola that is causing the nipples to project (which is often the case), and if so, how much of this glandular tissue can be removed to create a flat appearance.Also, men who suffer from large #areolas may notice that after breast surgery, the skin of the chest muscles will tighten, causing the areolas to tighten and diminish in size. This is a result of the removal of glandular and/or adipose tissue.

Thank you for your pictures. In my opinion you could benefit from God to Messi's surgery via liposuction. May require some glandular removal as well. I applaud your active lifestyle and I encourage you to continue it. 

There may be no local experience board certified plastic surgeons in your area but happily the problem is not that severe. Keep working on your exercise routine. At some point in your life if it's enough of a problem you will find your way to a plastic surgeon. Improvement is through a surgery which can be performed with a tiny incision inside your armpit leaving virtually no visible scar. Good luck with it.