It is often suggested that the only way to increase height is Growth Hormone or distraction osteogenesis surgery. However, there are way more methods to increase height than that. Just look at Michael Phelps, if there weren't local factors within the growth plate effecting growth then he would be perfectly proportional. Michael Phelps however is not perfectly proportional. If only HGH affected human height than people with disproportionate wingspans to leg length would not be a possibility.
One way to increase height is by increasing serum levels of cGMP(by means of Nitric Oxide and Guanyl Cyclase) or by inhibiting cGMP inhibitors(like PDE5 which is inhibited by Viagra). cGMP is related cGKII and cGKII knockout mice suffered from dwarfism. cGKII increases chondrocyte hypertrophy by promoting chondrocyte hypertrophic differentiation.
Chondrogenic hypertrophy is one of the final stages of endochondral ossification(chondrocyte hypertrophy is also influencable by IGF-1 of which serum levels can be increased by recombinant growth hormone treated cow milk and other dairy products). The other stages are the resting zone(the hyaline cartilage growth plate line), the proliferating zone, and the ossification zone(there may be ways to induce growth in the ossification zone but I have not learned of them yet).
In the resting zone, it is possible to increase height growth by way of getting new stem cells into the hyaline cartilage growth plate line. This can be achieved by hydrostatic pressure, pulsed electric magnetic fields, or low intensity pulsed ultrasound. You can also affect the DNA replicative capacity by altering telomere length or by altering DNA Methylation status. These are affected by telomerase and DNA Methyltransferase respectively. Human Growth Hormone may affect DNA Methylation which could explain it's role in Gigantism(HGH is involved in the feedback loop with IGF-1 as well).
You can also affect ion transport by pulsed electric magnetic fields. The Sodium Potassium pump has a profound effect on cell volume.
Then there's the proliferative zone. It is speculated that chondrocytes have a finite proliferative capacity but there is evidence of some flexibility. The optimal level of estrogen may set chondrogenic proliferative capacity. Myostatin(also known as GDF-8) is inhibited by testosterone. Myostatin doesn't only inhibit muscular cellular proliferation, it inhibits all cellular proliferation. Myostatin knockout mice were bigger and taller than the normal litter mates.
IGF-2, Lithium, and Puerarin may also affect height growth by their own mechanisms. IGF-2 is involved in an overgrowth disorder but it's exact mechanism of inducing height increase is unknown to me. Lithium increases stem cell proliferation but inhibits thyroid hormone. Puerarin is a PI3K pathway stimulator which is anabolic to all cells.
You can see that there are several potential scientific methods that hold promise in height increase. Unfortunately, most of them orginated from cancer research and male enhancement formulas than by direct height increase research. Support Height Increase research!