With VAR models you can predict as many steps ahead as you like. When the future horizon increases, you will eventually run out of observed values to base your forecasts on. Then you just substitute the required observed values with their forecasts and iterate forward. Here is an example of VAR(1) (I skip the intercept for brevity):
\begin{aligned}
x_t &= \Phi x_{t-1}+\varepsilon_t \\
\hat x_{t+1|t} &= \Phi x_{t} \quad \text{(have not run out of observed values yet)}\\
\hat x_{t+2|t} &= \Phi \hat x_{t+1} = \Phi^2 x_{t} \quad \text{(have run out of observed values, thus substitute)}\\
\hat x_{t+3|t} &= \Phi \hat x_{t+2} = \cdots = \Phi^3 x_{t} \\
&\cdots \\
\hat x_{t+h|t} &= \Phi \hat x_{t+h-1} = \cdots = \Phi^h x_{t} \\
\end{aligned}
where $x_t$ is a multivariate time series of interest, $\Phi$ is a square matrix of coefficients and $\varepsilon_t$ is a multivariate time series of errors (innovations, shocks).
Update to include some information from the comments:
We are doing the best we can given that we only have current information but not future information. The fact that forecasts are mean reverting reflects an assumption on the data generating process (DGP). As Chris Haug points out, this is a consequence of the linearity of the model, but it is not an approximation. The behavior of the forecasts is actually optimal given the DGP. And as long as VAR is a good approximation of the DGP, I would not expect alternative forecasts to beat the ones by VAR, unless they employ some future information. But in the latter case, a fair comparison would add this information to the VAR model in terms of exogenous variables or the like.
