Step 1: Remember the rule.
A function is differentiable on $\mathbb{R}$ only if it has no breaks and no sharp corners anywhere. Breaks or corners kill differentiability.
Step 2: Check $f(x) = [x]$.
The greatest integer function jumps at every integer, so it is not even continuous there, hence not differentiable.
Step 3: Check $f(x) = |x| + |x+1|$.
Absolute value graphs have sharp corners. This one has corners at $x = 0$ and $x = -1$, where the slope changes abruptly, so it is not differentiable everywhere.
Step 4: Check $f(x) = \frac{x}{|x|}$.
This equals $+1$ for $x > 0$ and $-1$ for $x < 0$, and is undefined at $x = 0$. It has a jump, so not differentiable on all of $\mathbb{R}$.
Step 5: Check $f(x) = \sin x + \log e^x + e^x$.
Note $\log e^x = x$, so this is $\sin x + x + e^x$. Each piece, $\sin x$, $x$, and $e^x$, is smooth and differentiable everywhere.
Step 6: Pick the smooth one.
The sum of three everywhere-differentiable functions is differentiable on $\mathbb{R}$. So this is the answer. \[ \boxed{f(x) = \sin x + \log e^x + e^x} \]