Is it grammatically acceptable to start a sentence with "That that..."

Solution 1:

It seems like this has been established in the comments, but in the interest of providing an answer, there's nothing ungrammatical about starting a sentence this way.

It's common to start a sentence with a pronoun and a determiner, as in this Confucius quote:

He who has really set his mind on virtue will do no evil.

Grammatically speaking, this is a variation on the same thing. Many writers, out of a belief that repeating a word is improper or just unpleasant, seem to substitute "which" for "that" in these situations. Ralph Waldo Emerson did so in the following quote:

That which we persist in doing becomes easier to do

But by most contemporary prescriptive rules for choosing between "that" and "which," that should be used when the restrictive clause alters the meaning of the sentence. Emerson wasn't merely adding the clause about persistence as additional information:

That, which we persist in doing, becomes easier to do

Contemporarily and prescriptively speaking, a more strictly adhering phrase would be:

That that we persist in doing becomes easier to do.

Just like a contemporary writer would generally prefer

Something that we persist in doing becomes easier to do

rather than

Something which we persist in doing becomes easier to do

But few people will judge Emerson for his stylistic choice of writing "that which," especially in poetry.

Another alternative, if you're interested in avoiding "that-that," is to use "what." "What" by definition can mean the thing or things that.

What is, is.

What we persist in doing becomes easier to do.

As pointed out in the question, using "that that" isn't foreign to talented writers. It is just as grammatical as the alternatives, and in many cases would be an appropriate thing to write. But if you're more interested in style and less interested in adhering to rules, you can rephrase sentences like this, and you'll be in the company of great writers either way.