Scale of variables and the consequences on the solution Let us say I have data set of distinct $x_i$. A Gaussian is fitted to it with maximum likelihood, obtaining some $\mu$ and some $\sigma^2$. I will also obtain a likelihood $\mathcal{L}$.
Now, I copy the data set where I set $\tilde{x}_i = 2 x_i$. I estimate another Gaussian, obtaining $\tilde{\mu}, \tilde{\sigma}^2, \tilde{L}$.
Since I am using ML I will get $2 \mu = \tilde{\mu}$, $4 \sigma^2 = \tilde{\sigma^2}$. Contrary to my intuition, $\mathcal{L} \neq \tilde{\mathcal{L}}$.
This is maybe not so bad for a univariate Gaussian. But what if I perform model selection for factor analysis, Gaussian mixture models or more complicated models? Will the order of models ranked by their respective likelihoods be the same?
 A: What dilemma?
While the actual value of the likelihood at the maximum will change, the point where it is maximized will be the same. 
E.g. let's take factor analysis:
First, from the help for factanal in R:
# A little demonstration, v2 is just v1 with noise,
# and same for v4 vs. v3 and v6 vs. v5
# Last four cases are there to add noise
# and introduce a positive manifold (g factor)
v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
m1 <- cbind(v1,v2,v3,v4,v5,v6)
cor(m1)
factanal(m1, factors = 3) # varimax is the default

Now, double all the variables and re-run
v1 <- 2*v1
v2 <- 2*v2
v3 <- 2*v3
v4 <- 2*v4
v5 <- 2*v5
v6 <- 2*v6

m2 <- cbind(v1,v2,v3,v4,v5,v6)

factanal(m2, factors = 3) # varimax is the default

The results are identical.  The actual value of the likelihood isn't relevant, only the place where it occurs. 
