How much total heat is contained in the upper layers of the atmosphere? In particular, I'm interested in the heat contained in the stratosphere, mesophere, and thermosphere.
Mass at 21.9.1998 for Height 140 km -1000 km was $6.16\cdot10^{10} \mathrm{kg}$.
Avg. Temperature; $725 \mathrm{K}$  

Roughly the mass is divided as follows;
$\mathrm{O}$, $\mathrm{O}_2$; $252 \cdot10^9 \mathrm{kg}$, “16” Molecule amount $N = 9.5\cdot10^{35}$,
Heat capacity; $920 \mathrm{J}/(\mathrm{kg} \mathrm{K})$

$\mathrm{N}$, $\mathrm{N}_2$; $364 \cdot10^9 \mathrm{kg}$; “14” Molecule amount $N = 1.56\cdot10^{36}$,
Heat capacity; $1040 \mathrm{J}/(\mathrm{kg}\mathrm{K})$

$\mathrm{He}$; $35.2\cdot10^6 \mathrm{kg}$, “4” Molecule amount $N = 5.3\cdot10^{33}$,
 Heat capacity; $5193 \mathrm{J}/(\mathrm{kg} \mathrm{K})$

$\mathrm{H}$; $1.34\cdot10^6 \mathrm{kg}$, “1” Molecule amount $N = 0.8\cdot10^{33}$,
Heat capacity; $14304 \mathrm{J}/(\mathrm{kg}\mathrm{K})$ 

Total Heat capacity; $61.2\cdot10^{12} \mathrm{J}/\mathrm{K}$ 
Quelle; My own work.

From here you can calculate $725 \mathrm{K} \cdot 61.2 \cdot10^{12} \mathrm{J}/\mathrm{K} =44.4 \cdot10^{15} \mathrm{J}$

Please note that most of the heat is at the lower atmosphere plain because >99% of the mass is there.