We can approximate the log likelihood at the real parameter value $l(\theta)$ with the MLE estimate $l(\hat\theta)$ using second order Taylor polynomials, like so:
$$l(\theta) - l(\hat\theta) \approx -\frac 1 2I(\hat\theta)(\theta - \hat\theta)^2$$
where $I(\hat\theta)$ is the observed Fisher information. It is always positive, as is the square term that follows. This implies that this difference in logs is negative. How can that be? This says that the log likelihood is higher under the MLE estimate than the true parameter value - shouldn't the truth be the highest attainable likelihood?