Should log-likelihood values increase when the sample size of a simulation increases? If one simulates a process (such as an ARMA-GARCH process) with sample size $n$ and log-likelihood $x$, should this log-likelihood increase when the sample size increases to $2n$ for example?
 A: It depends. More importantly though, it doesn't really matter. 
Remember, in an iid setting, the Likelihood is the product of PDFs (or PMFs) as a function of $\theta$. If each $f(x_i|\theta) < 1$ then the Likelihood will get smaller for each additional point. Uniform distributions make this point clear. 
Let $X_1, \cdots X_n \sim U(0, \theta)$, then clearly
$$L(\theta|{\bf X}) = \prod_{i=1}^n \frac{I(x_i < \theta)}{\theta} = \begin{cases}
\frac{1}{\theta^{n}} & ,\max(x_1, \cdots x_n) \leq \theta \\
0 & ,\max(x_1, \cdots x_n) >\theta
\end{cases}$$
or if you prefer
$$\log L(\theta|{\bf x}) = \begin{cases}
-n\log(\theta) & ,\max(x_1, \cdots x_n) \leq\theta \\
-\infty & ,\max(x_1, \cdots x_n) > \theta
\end{cases}$$
For values $\theta < 1$, the non-degenerate part of the likelihood gets arbitrarily big as $n$ increases. For value of $\theta > 1$, it gets arbitrarily small. But this doesn't really matter, since values of Likelihood aren't really interpretable. The important thing is that the Likelihood gets steeper around the likely values of $\theta$.
Here is a quick example.
Assume $X_1, X_2, \cdots X_n \stackrel{iid}{\sim} N(\mu, 1)$. Then
\begin{align*}L(\mu|{\bf X}) &= \left(\frac{1}{\sqrt{2\pi}}\right)^n\exp\left\{\frac{-1}{2}\sum_{i=1}^n(x_i - \mu)^2  \right\} \\
&= \left(\frac{1}{\sqrt{2\pi}}\right)^n\exp\left\{\frac{-\sum_{i=1}^n x_i^2}{2}\right\}\exp\left\{n\mu  (\bar{x} - \mu/2)\right\}
\end{align*}
For illustration purposes, we set $\bar{x} = 0$ and $\sum_{i=1}^n x_i^2 = n$ (this is what we "expect" if $\mu=0$).
Focusing on the vertical axis illustrates this point. The value of the likelihood itself doesn't really matter, it is the value of $L(\theta_1|{\bf x})$ relative to $L(\theta_2|{\bf x})$ that determines the likely values of $\theta$. 
