Is there a deep reason why strong estimates fail to exist for $L^1$ so often? For example, the Hardy-Littlewood maximal inequality fails to have a strong type estimate in the $(1,1)$ case.
Another example is the Hilbert transform which has strong type $(p,p)$ bounds for $1<p<\infty$ but fails to be bounded on $L^1$.
Is there a deep reason why such estimates fail so often for $L^1$? 
 A: I. There is a clear reason why bounding Fourier multipliers in $L^1$ is more difficult that doing so in $L^p$, with $1 < p < \infty$. $L^1$ is an Abelian Banach algebra with respect to the convolution and that Banach algebra, although it doesn't have a unit, it admits uniformly $L^1$-bounded approximate units (think of mollifications of $\delta_0$). The space of bounded Fourier multipliers in $L^1$ is equal to the so-called space of multipliers of the algebra $M(L^1)$ given by:
$$ 
  M(A) = \big\{ T \in B(A) : T(a \ast b) = a \ast T(b) \big\}.
$$
If the algebra $A$ had a unit $1$, then $T(f \ast 1) = f \ast T(1)$ and $M(A) = A$. Of course $L^1$ doesn't have a unit. But, using tricks with approximate units, it is not difficult to show that this space is given by convolutions with finite measures, ie: $T(f) = \mu \ast f$. A proof can be found in [St].
Therefore an $L^1$-bounded Fourier multiplier $m$ has to satisfy that $\widehat{m}$ is a finite measure. A pretty difficult thing that the Hilbert transform $$\xi \mapsto -i \operatorname{sgn}(\xi)$$ doesn't satisfy since its transform grow logarithmically.
II. For the maximals I do not have an "easy" reason. In the case of the Hardy-Littlewood maximal, if it were bounded in $L^1$ the strong maximal, given by
$$
  M_{\mathrm{st}}(f) := \sup_{r,s > 0} \, \sup_{|I| = r, \, |J| = s} \,  \frac1{r \cdot s} \int_{I \times J} |f(z,w)| dz \, dw,
$$
where $I$, $J$ are intervals such that $(x,y) \in I \times J$, would be bounded in $L^1$ as well. But it's very easy to see that that can not be, it is easier that in one dimension.
III. Another reason may came from the following paper of Stein [St2], that proves that $\| M(f) \|_1$ is comparable to the $L \log L$-norm using a Fubini-type trick.

[St]:Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton, N.J.: Princeton University Press. XIV, 287 p. (1970). ZBL0207.13501.
[St2]: Stein, Elias M., Note on the class $L \log L$, Stud. Math. 32, 305-310 (1969). ZBL0182.47803.
