In general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$; it is more of a request for guidance than a specific puzzle or conjecture. My present question has to do with an embedding of the infinite into a more familiar domain, namely, what (i think) Hardy referred to as a classification of real-valued functions by the rapidity with which they grow as their argument becomes very large. At the pre-university level this order is familiar, for example, in the degree of polynomials, or in the fact that the exponential functions grow faster than any positive power function, and that the latter class dominates the logarithms.
What do we see when we look a little more closely?
If we consider$$f_{\alpha}(x)=x^{\alpha} \; (\alpha \in \mathbb{R^+}) $$ then it is clear that$$ \bar f_{\alpha} \prec \bar f_{\beta} $$ whenever $\alpha \lt \beta$. Thus from a topological perspective the classes of power functions form an ensemble which is order-isomorphic to $\mathbb{R^+}$
This whole ensemble is dominated by classes of functions we may denote by $$ \bar f_{\alpha,\beta} = \alpha^{\beta x} $$ whose order is equivalent to the lexical ordering on $\mathbb{R}^+ \times \mathbb{R}^+$
This ensemble is again dominated by $$ \bar f_{\alpha,\beta,\gamma} = \alpha^{\beta x^{\gamma}} \; (\gamma \gt 1) $$ but dominates $$\bar f_{\alpha,\beta,\gamma} = \alpha^{\beta x^{\gamma}} \; (\gamma \lt 1). $$ it is not difficult to see that there is a whole family of such extensions of the power function, with an order topology isomorphic to the lexical ordering on $(\mathbb{R}^{+})^{\omega}$, but which family is dominated by the "2-parameter" family $$ \bar f_{\alpha,\beta} = x^{\alpha x^{\beta}}. $$
Now, it is not my purpose to continue elaborating these classes, or to discuss the extension of the same scheme in the downward direction beginning with the simple class of logarithms to different bases. I am interested in the bewildering scope of the order topology on these function classes, which i suspect, as hinted at in the title, bears some analogy with the long line offered as an interesting example in courses on elementary topology. I would be grateful for any soothing information which might help me to get a better night's sleep. I suspect that the most likely place i should look is recursive function theory, but i do not know much about this, and feel that its confinement to a finitistic domain will not necessarily assist with questions about the nature of the order topology on the full ensemble of function classes.