The genes governing inherited tendencies for taller or shorter stature are just beginning to be known. In an excellent article  about his own genome data, psychologist Stephen Pinker puts this in perspective:
"Though health and nutrition can affect stature, height is highly heritable: no one thinks that Kareem Abdul-Jabbar just ate more Wheaties growing up than Danny DeVito. Height should therefore be a target-rich area in the search for genes, and in 2007 a genomewide scan of nearly 16,000 people turned up a dozen of them. But these genes collectively accounted for just 2 percent of the variation in height, and a person who had most of the genes was barely an inch taller, on average, than a person who had few of them. If that’s the best we can do for height, which can be assessed with a tape measure, what can we expect for more elusive traits like intelligence or personality?"
And now back to the details:
A 2017 publication finds 83 height-associated coding variants with lower minor-allele frequencies (in the range of 0.1-4.8%) and effects of up to 2 centimetres per allele. The most significant (rare) variant was rs148833559, in the STC2 gene, with carriers 2cm taller than non-carriers; in contrast, carriers of the rare variants for rs137852591, rs148934412 or rs142036701 are 2cm shorter than non-carriers.[PMID 28146470]
nature SNP rs1042725 is associated with height (P = 4E-8) in a study involving over 20,000 individuals. The gene harboring this SNP, HMGA2, is a strong biological candidate for having an influence on height, since rare, severe mutations in this gene are known to alter body size in mice and humans.
Note that this SNP is by no means the whole story; rs1042725 is estimated to explain only 0.3% of population variation in height in both adults and children (approx 0.4 cm increased adult height per C allele), leaving over 99% of the influences on height to be described in the future ...
A study has been published in which 25,000 common SNPs explaining 42% of the variation in height (r2=0.65). Essentially all of the additive (linear) heritability might now have been found. Someone with the optimized genotype would be 15 feet tall. The study used a method called Lasso L1 penalized regression that could now be applied to other highly polygenic traits including intelligence. The potential implications of further research advances leading ultimately to the unlocking the human genome would be profound. 
Science Daily April 7, 2008: 20+ additional regions linked to variance in height based on 3 studies published in Nature Genetics (6 April 2008 Advance Online Publication)
nature 54 snps.
SNPs reported to be associated with height include:
- rs1867138 (T;T) or rs1867137(G;G) were approximately one inch taller than rs1867138(C;C)
- rs2179922 at least for males
- rs9834312(G;G) or rs939882(G;G) increased height
- rs9834312, in the FLNB gene, in both Caucasians and Chinese
- rs10734652, in the SBF2 gene, in both Caucasians and Chinese
- rs10816533, in Chinese populations
[PMID 28146470] Rare and low-frequency coding variants alter human adult height.
- Contains a list of ten height genes and 83 SNPs.
[PMID 19343178] Meta-analysis of genome-wide scans for human adult stature identifies novel Loci and associations with measures of skeletal frame size.
- Contains a list of 21 SNPs (see bold entries in Table 1 in this publication) replicating as associated with height in this 2009 study of ~20,000 individuals.