While I won't take any credit for this approximation as I don't know if someone else discovered it before myself, I did stumble upon this while playing around with the bound $ln(n)+\frac{1}{n} < H_n < 1+ln(n)$ for $n > 1$ which was mentioned above and the Euler-Mascheroni Constant.
Kudos to the person who first discovered this approximation (assuming there was one before myself) if they happen to see this post.
Approximation: $H_n \approx ln(n) + \frac{1}{n} + \gamma\left(1+ln\left(\frac{n}{n+1}\right)\right)$
where $\gamma = 0.577215664901532860606512...$ is the Euler-Mascheroni Constant.
Some calculations to back up this approximation
For $n = 100$
Actual Value: $H_{100} = 5.1873775...$
Approximation: $H_{100} \approx 5.1866423...$
Error: $-0.0007351...$
Percent Error: $0.0141719...$%
For $n = 1000$
Actual Value: $H_{1000} = 7.4854708...$
Approximation: $H_{1000} \approx 7.4853940...$
Error: $-0.0000768...$
Perent Error: $0.0010265...$%
I've also been able to stumble upon the fact that using the bound $ln(n+1) < H_n < 1+ln(n), n > 1$ (source: http://www.math.drexel.edu/~tolya/123_harmonic.pdf) allows you to get an approximation that is just a slight bit more inaccurate than the first one I presented at the top.
Approximation: $H_n \approx ln(n+1) + \gamma\left(1+ln\left(\frac{n}{n+1}\right)\right)$
This approximation is simpler, and a slight bit easier to calculate. However, it is slightly less accurate, so unless you are desiring extreme precision, this one may be more appealing.
The above two approximations are is relatively easy to calculate compared to some others you may find out there (like Ramanujan).
A very simplified form of Ramanujan's approximation is
$H_n \approx ln(n) + \gamma$
The above two approximations I presented are both quite a bit more accurate than this truly oversimplified version of Ramanujan's approximation. As @Winther pointed out, there is an error of approximately $\frac{0.077}{n}$ which implies that (just like pretty much all other approximations) the approximations are much closer to the actual value as $n$ gets larger. However, Ramanujan's approximation in its complete form is extremely accurate, though it is extremely complex.
If anyone wishes to know the motivation behind the derivation of these approximations (at least from how I did it), then I am happy to answer in the comments.
EDIT: I stumbled upon one that is more accurate, again kudos to whoever may have discovered it before me. Also, this is more accurate than the approximation $ln(n)+\gamma+\frac{1}{2n}$ up to some point.
$H_n = ln(n) + \gamma\left(1+\frac{50}{51n}+ln\left(\frac{n-\frac{\gamma}{10}}{n+\frac{\gamma}{10}}\right)\right)$