LM317 voltage regulator with high current boost by 3 npn transistors
Although I've not specifically attempted a \$15\:\text{A}\$ boosted LM317 before, this is along the lines of what I'd try out first. This is roughly taken from the Figure 23 you mentioned:







simulate this circuit &ndash; Schematic created using CircuitLab

In this case, I went for the D44/D45 series devices. (The PNP version has simply HORRIBLE Early Effect, but it's not a big deal here.)

The values of \$R_6\$, \$R_8\$, and \$R_9\$ are set to drop somewhere from \$150-200\:\text{mV}\$ at full load. They will need to be rated for at least \$1\:\text{W}\$, but I would not feel comfortable with less than \$2\:\text{W}\$ resistors there. If you adjust those values, please keep in mind the dissipation question. You are talking about a lot of current.

To reduce the oscillation, you really want some ESR in \$C_2\$ to add a nice 'zero'. If you see oscillation in the output, try adding a small series resistor to \$C_2\$. \$15-39\:\text{m}\Omega\$ (as shown with \$R_{10}\$) should put a crimp in the oscillation. You might just make provisions for it and jumper it, without using a resistor, if your output seems fine with the output capacitor you selected. But here is one of those cases where output capacitor ESR is actually a good thing.



Your schematic shows an AC input. That's not good. I hope your schematic was just mistaken, there.



Since the minimum specification for the LM317 is \$3\:\text{V}\$ from input terminal to output terminal, the externally added circuit will always have more than enough headroom to operate so long as you supply that difference.

Keep in mind this is a linear power supply. With \$\approx 3.3\:\text{V}\$ output and \$\approx 3\:\text{V}\$ overhead, you will have little better than 50% efficiency. At full load, you will have \$\ge 45\:\text{W}\$ wasted dissipation, not counting the load's dissipation. And more than that, likely, because this ignores whatever you have supplying the unregulated input DC voltage -- where it is likely you have still more dissipation in diode rectifiers from AC, etc.

While perhaps \$3\:\text{W}\$ dissipation might occur in the emitter resistors, that still leaves a pretty much all the rest with the bypass BJTs. Getting rid of \$15\:\text{W}\$ each will be the challenge. Note that if you want to allow a maximum junction temperature of say, \$100\:^\circ\text{C}\$, and the worst case ambient temperature you care about supporting is \$45\:^\circ\text{C}\$, then this means you need \$\frac{100^\circ\:\text{C}-45^\circ\:\text{C}}{15\:\text{W}}\approx 3.7\:\frac{^\circ\text{C}}{\text{W}}\$. For the parts I mentioned, junction to case is already \$1.8\:\frac{^\circ\text{C}}{\text{W}}\$. That leaves you only \$1.9\:\frac{^\circ\text{C}}{\text{W}}\$ for whatever you use as a heatsink plus the bonding interface between the BJTs and that heatsink. That's not a lot to work with.

You might consider putting more of the dissipation into the emitter resistors, I suppose. More degeneration won't hurt you. I chose to set them at about a minimum resistance for the circuit, so increasing their values will be fine. (Don't decrease them much, though.) You need to work out this balancing act on your own.