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What can't be expressed as a linear model?

Say I have outcome variable $Y_i$ and predictors $X_{i1}$ and $X_{i2}$ for some data point $i$. Wikipedia says that a model is linear when:

the mean of the response variable is a linear combination of the parameters (regression coefficients) and the predictor variables.

I thought this meant that a model can be no more complicated than: $Y_i = \beta_1 X_{i1} + \beta_2 X_{i2}$. However, upon further reading, I found out you could handle non-linear "interactions" of the predictors like in $Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X_{i1}\ X_{i2}$ by viewing $X_{i1}\ X_{i2}$ as just another predictor (which happens to be dependent on $X_{i1}$ and $X_{i2}$). This seems to mean that you can use any (linear or non-linear) function of the predictors like $\log(X_{i1} / X_{i2}^2)$ or whatever. Conceptually, this type of "recoding" seems like it should work for the coefficients as well.

So: What exactly are the limits of linear regression, given you can do this kind of manipulation?