Can I Get a Natural Breast Enhancement with Fat Injections?

There is a company called Cytori that is conducting clinical trials of breast augmentation overseas using your stem cells, but it doesn't quite work like you described.
The plastic surgeon harvests fat cells, just like with liposuction. They are processed over about an hour, and the stem cells are separated out. They add these concentrated stems cells to the fat that they are going inject to "turbocharge" the injection, so that more of the fat cells stay. One of the problems with fat injections is that much of it gets resorbed by the body, but this appears to help the process. I have spoken to several of the doctors doing this, and they say that it is promising, but that they have only been doing it for a few months and that it is too early to tell how well it really works.
But it is only being done in clinical trials, and it is far from being recommended for routine use. And so far, it appears helpful only in small degrees of augmentation, such as less than a cup.
Injecting anything into the breast, particularly stem cells, must be done with great caution, and it should not be recommended for general use until there is a long track record of experience. That is because the breast is a very cancer-prone organ, and most women don't reach the age that cancer is common until several decades or more after getting some sort of an augmentation. We know that regular breast implants - silicone and saline - do not increase the chance of breast augmentation one iota. It will take a long time before we know that about injections of stem cells and fat into the breast.
It sounds great. And it will be. It will undoubtedly come sooner than "phasers' , but it is still a very long way off.

Fat transfer may be natural and sound like a good idea, there are many problems with it:1. only one cup size can be achieved2. some of the re-injected fat calcifies, which could be mistaken for breast cancer on a mammogram.3. you may develop oil cysts and tissue scarring4. results can be unpredictable and require many sessions to achieve your desired results.Currently, there is no research that proves fat transfer is safer than implants (silicone or saline).

As of right now, breast enhancement with fat cells, or using your own harvested fat, is not generally recommended. The reason is that with any fat transfer, you never quite know how much of the fat is going to live and how much is going to die off.
If some of the fat dies off ( animal studies show that about 50% will be gone in a year), then it can lead to calcifications, mimicking what breast cancer clusters can look like ona mammogram. This can lead to false positives on a mammogram, or mask detection of breast cancer calcifications. Until there is a reliable technique to harvest and transfer fat that results in 100% viability over time without calcification, it will always be contraindicated.