Uncountable set of irrational numbers closed under addition and multiplication?

This is possible.

First, consider the set of all numbers of the form $$ a_1\pi + a_2\pi^2 + \cdots + a_n\pi^n $$ where $n \geq 1$, the coefficients $a_1,\ldots,a_n$ are non-negative integers, and at least one $a_i$ is positive. This set is clearly closed under both addition and multiplication. However, it is not uncountable.

We can make this set larger by adding another number. For example, we can consider two-variable polynomials involving $\pi$ and $e$ with the same restrictions: there is no constant term, all of the coefficients are non-negative integers, and at least one of the coefficients is positive. Assuming that $\pi$ and $e$ are algebraically independent (which is not known), all of these polynomials are distinct and nonzero, so we get a larger set of transcendental numbers which is closed under addition and multiplication. However, this set is still not uncountable.

To make an uncountable set that is closed under addition and multiplication, we must start with an uncountable set $S$ of algebraically independent transcendental real numbers. It is known that such a set exists, since the transcendence degree of the real numbers over the rationals is uncountable. If we now take all polynomials over the elements of $S$ satisfying the same conditions (no constant term, non-negative integer coefficients, at least one positive coefficient), the result will be an uncountable set of transcendental numbers that is closed under addition and multiplication.

Let $A_0$ be a transcendence basis for $\mathbb{R}$ over $\mathbb{Q}$, which already has size of the continuum, and let $$B_0=\{\text{finite sums of elements of }A_0\},$$ $$A_1=\{\text{finite products of elements of }B_0\},$$ $$B_1=\{\text{finite sums of elements of }A_1\},$$ $$\cdots$$ and $S=\bigcup_{n=0}^\infty A_n$. Then $S$ is closed under addition and multiplication by definition, $S$ is uncountable, and every element of $S$ is irrational because no algebraic relation holds between the elements of $A_0$ over $\mathbb{Q}$.

EDIT: This is the same as Jim's suggestion above, just phrased slightly differently.

To extend the comment above.

The property: "$S\subset \mathbb R\setminus\mathbb Q$ is closed under addition and multiplication" satisfies Zorn's lemma, so there must be a maximal such set $S$.

Maximality means that for any irrational number $x\notin S$, there must be a polynomial of the form $p(z)-q$ where $q\in \mathbb Q$ and $p(z)$ is a non-zero non-constant polynomial with coefficients in $(S\cup\{0\})\oplus\mathbb N)$ such that $x$ is a root of the polynomial.

But if $S$ is countable, then the set of such polynomials is countable, and therefore the set of such roots is countable.

So any maximal $S$ must necessarily be uncountable.

This can be extended to a more general theorem: If $K\subset \mathbb R$ is uncountable and closed under addition and multiplication, then there is an uncountable $S\subseteq K\setminus\mathbb Q$ which is closed under addition and multiplication.

Proof: Since $K$ is uncountable, it contains a transcendental number, so the positive polynomials in that transcendental form a subset of $K\setminus\mathbb Q$ closed under addition and multiplication.

As before, by Zorn's lemma, there must be a maximal set $S\subset K\setminus\mathbb Q$ closed, and, as before, if $S$ is countable, then you get a contradiction - there are too many elements of $K\setminus (S\cup\mathbb Q)$ and not enough polynomials with coefficients in $(S\cup\{0\})\oplus\mathbb N$.

Here's a more "consructive" answer, without using Axiom of Choice. Let ${\cal F}_n$ be the set of functions of $n$ variables corresponding to expressions using each variable once, with $+$ and $*$ (a convenient way to represent these is as trees where each leaf is one of the $n$ variables and each non-leaf node is labelled with either $+$ or $*$: in any case, there are finitely many for each $n$, and they can be enumerated). Note that ${\cal F}_1$ consists of the one function $x \to x$. I will take $S = \bigcup_{n=1}^\infty \bigcup_{f \in {\cal F}_n} f(C, \ldots,C)$ for a suitable uncountable set $C$. I need to ensure that for all $f \in {\cal F}_n$, $f(C,\ldots,C)$ does not contain any algebraic numbers.

Enumerate the triples $(\alpha, n, f)$ where $\alpha$ is algebraic, $n$ is a positive integer and $f \in{\cal F}_n$, as $(\alpha_k, n_k, f_k)_{k=1}^\infty$. $C$ will be obtained as a nested intersection $\bigcap_{k=0}^\infty C_k$, where $C_0 = [0,1]$, each $C_k$ is the union of $2^k$ disjoint closed intervals, and each interval of $C_{k-1}$ contains two intervals of $C_{k}$. These are to be chosen so that $\alpha_k \notin f_k(C_{k},\ldots,C_k)$. To do this, note that if we randomly choose $x_i$ and $y_i$ in each interval of $C_{k-1}$, with probability $1$ none of the numbers $f(z_1,\ldots, z_{n_k})$ for $z_1, \ldots, z_{n_k} \in \{x_i: i=1..2^{k-1} \} \cup \{y_i: i =1..2^{k-1}\}$ is $\alpha_k$. Taking $\epsilon > 0$ small enough, we can let $C_k$ be the union of the intervals $[x_i - \epsilon, x_i + \epsilon]$ and $[y_i - \epsilon, y_i + \epsilon]$ for $i=1 \ldots 2^{k-1}$, and this will have the required properties.