How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$? Let $\zeta$ be the Riemann zeta function.
My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ? 
My searches in relevant texts like Titschmarsh reveal that people seem to be focused on the case $\sigma= 1/2$, for which Lindelof conjectured that $|\zeta(1/2+it)|\ll t^{\epsilon}$ for any $\epsilon>0$.
 A: I answer here for what is known for general $\sigma$, without assuming anything (in particular not LH). Let $s = \sigma + it$ where $s$ and $t$ are real numbers.
If $\mathbf{\sigma > 1}$, then the absolute convergence of the Dirichlet series yields
$$\zeta(s) \ll 1 \qquad(\sigma>1)$$
(here and below, the $\ll$-constant may depend on $\sigma$.
If $\mathbf{\sigma < 0}$, then we can use the functional equation in order to come back to the known region $\sigma > 1$. Indeed, recall that $\zeta(s) = \gamma(s) \zeta(1-s)$ for a certain explicit completing factor $\gamma(s)$. It can be written in terms of $\Gamma$ functions and the Stirling formula easily leads to
$$\gamma(s) \ll_\varepsilon |t|^{\frac{1}{2} - \sigma + \varepsilon}$$
Since $\zeta(1-s)$ is bounded for $\sigma < 0$ by the previous case, we deduce
$$\zeta(s) \ll |t|^{\frac{1}{2} - \sigma} \qquad(\sigma<0)$$
If $\mathbf{0 \leqslant \sigma \leqslant 1}$, the Phragmén-Lindelöf principle allows to interpolate the bounds above in the critical strip, so that
$$\zeta(s) \ll |t|^{\frac{1}{2}(1 - \sigma)}\qquad(0<\sigma<1)$$
Good references for such matters are for instance Montgomery-Vaughan, Multiplicative Number Theory, 1. Classical Theory, or Tenenbaum, Introduction to Analytic and Probabilistic Number Theory.
As for the Lindelöf hypothesis (implied by RH, so this bound holds under RH as well.), it states as you said that for $\sigma = 1/2$ a certain bound is satisfied, namely
$$\zeta(s) \ll_\varepsilon |t|^\varepsilon.$$
Is you are interested in what happens in the whole critical strip, you can apply the Phragmén-Lindelöf principle as well to sharpen the bound. You get assuming LH (or RH),
\begin{align*}
\zeta(s) &\ll_\varepsilon |t|^{\varepsilon}\qquad\left(\frac12\le\sigma<1\right) \\
\zeta(s) &\ll_\varepsilon |t|^{\frac12-\sigma+\varepsilon}\qquad\left(0<\sigma<\frac12\right)
\end{align*}
