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The value 8 is not a solution, but there is a solution whose approximate solution is 8.2510146...

Here are some ways to work on solving this equation.

Define t as a symbolic variable in the symbolic ring, and call eq the equation.

sage: t = SR.var('t')
sage: eq = 0.111*t == 1-exp(-0.3*t)

Try to solve with solve: unfortunately, Sage returns an equation which is equivalent to the equation we started with.

sage: solve(eq, t)
[t == 1000/111*(e^(3/10*t) - 1)*e^(-3/10*t)]

Use find_root to find an approximate solution between 5 and 10.

sage: eq.find_root(5, 10)
8.251014632362164

or

sage: find_root(eq, 5, 10)
8.251014632362164

The computation above is done using floating-point computations and it is not clear which digits are exact.

For a computation using arbitrary precision, use the mpmath library: define a function equal to the difference of the left-hand side and the right-hand side of eq and use mpmath.findroot to look for a root near 8:

sage: import mpmath
sage: mpmath.findroot(lambda t: 1 - exp(-0.3*t) - 0.111*t, 8)
mpf('8.2510146323620207')

See the answer by @Emmanuel Charpentier for how to use SymPy to get an exact solution in symbolic form. It is likely Giac or FriCAS could do it too.