I have derived the following alternating infinite series as an approximation to values of the Gamma Function which yields reasonably accurate results between Gamma(1) and Gamma(7)-ish. Beyond that, the approximations oscillate and increasingly diverge from the true Gamma values. Do you see any opportunities to rein in this series’ unruly behavior?
EDIT: Another path of inquiry has arisen. Since posting this, I’ve entered my formula into a spreadsheet which calculates the sum of the first 170 terms of the series (the iPad spreadsheet app will only calculate up to 170!). It now occurs to me that perhaps the deviations I’ve observed for large m might indeed be accounted for by either accumulated rounding error or some other deficiency in the app’s underlying calculation methods.
In my spreadsheet, sqrt(exp(e + pi)) is rounded (or truncated, perhaps) to 18.7264552066483. The intriguing thing is that, for large m, when I make a change of just 1 unit in the 13th and last decimal place, this radically changes the resulting sum and hence, the percent error of my approximation. Now this might be wishful thinking, but if the series is that utterly sensitive to minuscule changes, perhaps the mounting errors for large m are an artifact of inadequate precision in processing the calculations.
$$\displaystyle \Gamma(m) \approx \sum_{n=0}^{\infty}(-1)^n \frac{\left(\sqrt{e^{e+\pi}}\right)^{m+n}}{n!(m+n)}$$
To simplify the notation, I define $$g = \exp((e+\pi) / 2)$$ $$g\approx$$
18.72645520664827426714604454007043332367490879335810853397969311801774145367to 256 bits of precision.
Your series can be written as $$\Gamma(x)\approx \frac{g^x}x -\frac{g^{x+1}}{1!(x+1)} + \frac{g^{x+2}}{2!(x+2)} - \frac{g^{x+3}}{3!(x+3)} + \ldots $$
Here's some Sage code that calculates that series, to any desired precision. (SageMath is a free open-source mathematics software system based on Python and built on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R).
Here's the output for the default inputs of $m=4$, $N=100$ terms, with 128 bits of precision used for the calculations. (Standard floating point arithmetic on most modern systems has 53 bits of precision).
Using the SageMath RealField class for the arithmetic, the series appears to be quite stable, once $N$ is large enough, even at relatively low precision.
Here's a live link to the script, running on the SageMathCell server, so you can run it in your browser, even on an iPad or phone.
FWIW, Sage gives this expression for the sum of your series:
$$\frac{{\left({\left(x + 1\right)} e^{\left(\frac{1}{2} \, \pi x + \frac{1}{2} \, x e\right)} \gamma\left(x + 2, -e^{\left(\frac{1}{2} \, \pi + \frac{1}{2} \, e\right)}\right) - {\left(x^{2} + x\right)} e^{\left(\frac{1}{2} \, \pi x + \frac{1}{2} \, x e\right)} \gamma\left(x + 1, -e^{\left(\frac{1}{2} \, \pi + \frac{1}{2} \, e\right)}\right)\right)} e^{\left(-\frac{1}{2} \, \pi - \frac{1}{2} \, e\right)}}{x^{2} \left(-e^{\left(\frac{1}{2} \, \pi + \frac{1}{2} \, e\right)}\right)^{x}}$$
where $$\gamma(a,z)=\int_0^z t^{a-1}e^{-t}\,\mathrm{d}t$$ is the lower incomplete gamma function
It turns out that the expression for the sum of your series can be simplified to $$\gamma(x, g)$$
I found that by telling Sage to use a variable for $g$ rather than its symbolic expression in terms of $\pi$ and $e$.
live link
And here's an updated version of the main script which prints the value calculated using the lower incomplete gamma function ("gil") before it starts summing the series. This value doesn't suffer from the loss of precision caused by the large terms in your series, so its value differs from the sum of the series in the last 10 digits when calculating the series for $m=4$.
Here's a plot of $\Gamma(x)-\gamma (x,g)$ over $[0,4]$
created using this plotting script, which can plot the difference over any specified domain.
The Wikipedia article on the incomplete gamma functions gives generalised continued fractions for both the upper and lower incomplete gamma functions. This script calculates both functions using those continued fractions, and sums them to give an estimate of gamma.