In some sense, is linear regression an estimate of an estimate of an estimate? Consider the problem of estimating a random variable $Y$ using another random variable $X$.
The best estimator of $Y$ by a function of $X$ is the conditional expectation $E[Y|X]$. It minimizes the mean-squared error $E[(Y - h(X))^2]$ over all $L^2$-functions $h(X)$ of $X$.
However, $E[Y|X]$ seldom has a nice closed expression. A further approximation is to consider the best linear unbiased estimator
$$ linear[Y|X] = \mu_Y + \rho \frac{\sigma_Y}{\sigma_X} (X - \mu_X) $$
However, in practice we don't even know the population variances and population correlation. So we use linear regression techniques such as the least-squared solution to get an estimate from a random sample.
In this sense, can we say that linear regression is an estimate of an estimate ($linear[Y|X]$) of an estimate ($E[Y|X]$) of the original random variable ($Y$)?

A frequentist would use the term "prediction" for what you call "estimation of a random variable". Furthermore, if you don't assume a linear relationship but just use this as approximation, using the term "estmation" for "approximation" is confusing. And even then what is "estimated" is the coefficients of the regression, not the prediction of $Y$. So you're estimating the coefficients of an approximation in order to predict $Y$.
Things are not so clear in Bayesian analysis where parameters to be estimated are also random variables. There $E[Y|X]$ may be legitimately called "estimator" (even though some would probably still see this as confusing use of terminology), and you may estimate involved regression parameters on top of it. I don't see any way how to justify the term "estimator" for the linear approximation though in a Bayesian framework.
