Summary: We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of \(2\pi\)-periodic functions whose \((\psi,\beta)\)-derivatives belong to unit balls in the spaces \(L_p\), \(1\leq p < \infty\), in the case where the sequence \(\psi(k)\) is such that the product \(\psi(n)n^{1/p}\) may tend to zero slower than any power function and \(\sum_{k=1}^{\infty} \psi^{p'}(k)^{p'-2}<\infty\) for \(1<p<\infty\), \(\frac{1}{p}+\frac{1}{p'}=1\), or \(\sum_{k=1}^{\infty} \psi(k)<\infty \) for \(p = 1\). Similar estimates are also established in the \(L_s\)-metrics, \(1 < s \leq \infty\), for the classes of summable \((\psi, \beta)\)-differentiable functions such that \(\|f_\beta^\psi\|_1\leq 1\).