Existence/Uniqueness of solutions to quasi-Lipschitz ODEs Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz in y?
If not, are there any moduli of uniform continuity weaker than Lipschitz continuity that it is known suffice, or results indicating that there can't be any?
 A: Yes. This follows from the classical uniqueness theorem due to Osgood (the original paper appeared in 1898).

Osgood's Criterion. Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous
on the interval $(0,a)$ and $\psi(u)$ is continuous on $\mathbb R_{+}$, $\psi(0)=0$, $\psi(u)>0$ for $u>0$, and
$$\int_{0}^{\epsilon}\phi(t)dt<\infty,\qquad \int_{0}^{\epsilon}\frac{du}{\psi (u)}=\infty$$
for some $\epsilon>0$. Suppose that the mapping $f:[0,a]\times B_R(x_0)\to \mathbb R^d$ satisfies the condition
$$\|f(t,x_1)-f(t,x_2)\|\leq\omega (t,\|x_1-x_2\|)$$
for any $t\in(0,a]$ and any $x_1,x_2\in B_R(x_0)$. Then the initial value problem
$$\dot{x}=f(t,x),\qquad x(0)=x_0$$
has at most  one solution on the interval $[0,\delta]$ with some $\delta>0$.

Osgood's theorem allows for the mappings $f(t,x)$ which are discontinuous at $t=0$. (Actually, the condition that $\phi(t)$ is continuous on $(0,a)$ can be replaced with an assumption
of mere integrability.)  Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that $f$ is continuous in $(t,x)$.
Moreover, Wintner showed that Osgood's uniqueness condition implies the  convergence of  successive Picard iterations to a local solution on a sufficiently small interval (A. Wintner, "On the Convergence of Successive Approximations", Amer. Journal of Math. Vol. 68 (1946), pp. 13-19).
