Problem. Integration of function a^x stop and gives error-warning: "Result is above max. allowed positive number" ( ~ 10^+310 ).

Question. Is it possible to increase the maximum allowed number by programme setting?

Unfortunately, no. This is the 64-bit IEEE float upper limit, and SMath uses this as its number format. There's no extended precision number support in SMath at the moment.

For big number integration, Smath is limited to e-307
There used to be an on-line Mathematica integrator ...
for big numbers, but useless for an Smath suite.