I've been trying to answer this question for several years and it turned out to be really  hard, even for the $2$-sphere.   Below I will discuss this case. 

First of all  one should  ask   what  is the number $m(k)$ of topological types  of (stable) Morse functions on $S^2$  with  precisely $k$ saddle points. (such a function has $2k+2$ critical points.)  I showed that the generating series

$$ f(t) := \sum_{k\geq 0} \frac{m(k)}{(2k+1)!} t^{2k+1}, $$

is the inverse  of an elliptic integral; see  [this paper][1]. This fact   leads to a positive answer to a question of V.I. Arnold   who conjectured that  $\log m(k)\sim 2k\log k$ as $k\to \infty $. I refer you to  [this paper][2] for details.   This shows that $m(k)$ grows rather fast as $k\to \infty$.

Any   polynomial $P$ of degree $d$ in $\newcommand{\bR}{\mathbb{R}}$ on $\bR^n$   can  be uniquely decomposed  as a sum

$$ P= \sum_{0\leq j+2k\leq d} r^{2k} H_{j}, \;\; r^2= (x_1^2+\cdots +x_n^2), $$

where $H_{j}$ is a darmonic polynomial of degree  $j$.      On $\bR^3$ the space of degree $d$ hormonic polynomials has dimension  $2d+1$.   If we denote by $U_d$ the subspace of $C^\infty(S^2)$ consisting of the restrictions to $S^2$   of the  polynomials of degree $\leq d$  we deduce that

$$\dim U_d=\sum_{0\leq k\leq d} (2k+1)=(d+1)^2. $$

I showed  that  as $d\to \infty$ the expected number of critical points of a function in  $U_d$  is  comparable to $C \dim U_d\sim  C'd^2$,  where $C, C'$  are certain explicit constants; see [this paper][3] and [this paper][4].  Denote by $K_d$ this expected number. Moreover it turns out that the   number of critical points of a random function in $U_d$ is     highly concentrated around its mean $K_d$, i.e., the probability that a random function  in $U_d$  a number of critical far from the mean  $K_d$ is extremely small as $d\to\infty$.   

I personally believe, based on some empirical evidence,   that the mean  is close to the  maximum number of critical points    in the sense  that if  we denote by $\mu_d$ the maximum number of critical points of a  Morse function in $%U_d$ then $\mu_d \sim C'' d^2$ as $d\to\infty$.

 My guess is that the number of topological types of functions in $U_d$  as $d\to \infty$ is roughly

$$ \sum_{k=1}^{K_d/2} m(k), $$
where $m(k)$  I recall that $m(k)$ denotes the number of topological types  of Morse functions with $k$ saddle points, i.e., $2k+2$ critical points.




  [1]: http://www3.nd.edu/~lnicolae/Morse-count.pdf
  [2]: http://www3.nd.edu/~lnicolae/statistics.pdf
  [3]: http://front.math.ucdavis.edu/1008.5085
  [4]: http://www3.nd.edu/~lnicolae/CritSetStat.pdf