TL;DR: If by “analogue computers”, you mean differential analysers, the answer is adders, constant units and integrator. Bournez, Campagnolo, Graça and Hainry have shown in 2006 (paywalled / free reprint) that an idealized model of it allows to compute all the computable functions in the framework of computable analysis, and this model only needs these 3 kind of units.
The set of operation you propose (addition, multiplication, subtraction, and division), even completed by root equation is by definition not sufficient to compute any transcendental function. Transcendental functions includes very common functions, like $\sin$, $\exp$, $\log$. However, as discussed below, some analogue computer model allow to compute transcendental functions and, basically all real functions computable by a Turing machine.
Analogue computing models
As stressed by others, the concept of “universal computation” is less clear for analogue computers than for standard computer, where different natural notion of computability in different computing models where found equivalent in the 1930’s (see Wikipedia page on Church Turing Thesis for details).
In order to define such a universality, one should first define a good model for analogue computation, and it is a difficult task, since the model should be idealized and and natural enough to be useful, but its idealization should not give unrealistic power to the model. An example of such a good idealization is the infinite tape of Turing machines. The problem with analogue computers comes with real numbers which could allow to build unreasonable stuff like the Zeno machine. However, several such models have been proposed and used in the literature (The GPAC is the main subject of this answer, but I try to be complete in the list below, without any hypercomputer):
Power of the GPAC model
In his 1941 paper, Shanon introduced the GPAC to model differential analysers.This model only needs 3 kinds of interconnected units (constant units, adders and integrators. Multipliers can be built from integrators and adders.)
He showed that the set of functions which it generates is the set of algebraic differential functions, but excludes the hypertranscendental functions. It means that the $\Gamma$ and $\zeta$, which are Turing-computable cannot be generated. In other words, no differential analyser will ever have an output $y(t)=\Gamma(t)$, ant it seemed for a long time that such an analogue computer is not “universal”, since it cannot generate some reasonable computable functions, used by mathematicians.
However, in 2004, Daniel Silva Graça showed that the previous model, based on instantaneous computation is too restrictive. If one define the computability of a function $f$ differently, allowing $y(t)$ to converge towards $f(x)$, for an input $x$, then the $\gamma$ and $\zeta$ functions are computable by a GPAC. Bournez, Campagnolo, Graça and Hainry then showed in 2006 (paywalled / free reprint) that an idealized model of it allows to compute all the computable functions in the framework of computable analysis.
Bournez, Graça and Pouly then showed in 2013 that these analogue computers can efficiently simulate a Turing machine (p.181 of a big pdf) and, in 2014, that the P and NP complexity classes are equivalent in this model.