Although, the function e^{t^2} is not exponentially bounded and due to linearity of Laplace transform we may write
L {(e^{t^2}}=L{1+t^2+(t^4)/2!+(t^6)/3!+....}
=1/s+(2!)/s^3+(4!)/(2!s^5)+(6!)/(3!s^7)+.....
By using power series of e^{t^2}, perhaps, L {(e^{t^2}} can be computed.