Have any Christian thinkers expressed the idea that the Trinity can be proven independently from Scripture?

Anselm of Canterbury attempts to do exactly this in his Monologion, an 11th century work that attempts to logically deduce God's existence and attributes without the use of Scripture.

Overview in Proslogion

Anselm's later and better-known work, Proslogion, provides a helpful summary: after establishing the existence of God through the ontological argument (§2–3) and that God cannot be divided (§18), Anselm introduces a Father/Son relationship between God and a new entity, God's Word. This Word shares God's attributes, like truth, and thus is not distinct from God. A third entity, God's Love, is named the Holy Spirit, and this Love is "one and common" to God and God's Son, and equal to both. Summarizing, he writes:

Thus, whatever each is singly, that the whole Trinity is altogether, Father, Son, and Holy Spirit; since each singly is not other than the supremely simple unity and the supremely unified simplicity which can be neither multiplied nor differentiated. (§23)

Details in Monologion

The Son

The Monologion provides more extensive reasoning for the same basic argument as the Proslogion. The existence and characteristics of the supreme nature are argued first, and then in chapter 29, Anselm turns to the "verbalization" of the supreme nature, "through which all things were created." He argues that the verbalization is same as the supreme nature, and that its verbalization is consubstantial with it, so that there is only one spirit, not two.

Because of the principle of simplicity, established in chapter 17, the verbalization, or Word, is one, and not many (§30). The supreme spirit creates everything through this Word, and "whatever the supreme spirit does, its Word does too, and likewise." (§37). The supreme spirit begets the Word, and the Word "derives from the supreme spirit" (§39), like a Father/Son relationship (§42). Nonetheless, "there are not two, but only one Creator and first principle." (§37) But this raises the question, then: of what, exactly, are there "two"?

Anselm answers that this plurality is "ineffable," incapable of being expressed in words. They are not like "two similar men," nor are they "two equal spirits," nor even "two equal whatevers (where 'whatever' would signify either their essence or their relation to creation)." Finally, he concludes:

What these two things—the supreme spirit and its Word—are, is inexpressible, although they are distinguished from each other in thought by what is proper to each. (§38)

The Holy Spirit

Following Augustine (De Trinitate, IX–4), Anselm turns to a psychological analogy to describe these two entities, and associates consciousness with the Father, and understanding with the Son (§48). But one thing is missing: just as the supreme spirit is conscious of itself, and understands itself, it must also love itself. This love, proceeding from both the self-consciousness (Father) and the self-understanding (Son), is as great as both of them (§50–52).

Love, then, is the Spirit of the Father and of the Son (§57). Unlike the Word, which naturally "offers the image of its parent," Love provides no "striking likeness," so we can only say that it proceeds from Father and Son (§55). Thus, the Father is unbegotten and begets the Son; the Son is begotten; and the Spirit is neither begotten nor unbegotten (§56).

These three are equal, sharing the one supreme essence, and their "mutual embrace" and "indwelling [...] demonstrates that none of them is greater than, or can exist without, the others." (§59) Nonetheless, Anselm writes, "none of them needs the others in order to be conscious, understand and love." Otherwise, each individual would not truly be "supreme essence and wisdom." (§60)

After a few rebuttals to objections, Anselm begins his conclusion of this portion of the argument by writing:

This seems to me to be a sublime mystery, which stretches well beyond the horizon of human understanding. [...] When investigating the inexplicable, if it is possible to arrive at an account which is certainly correct, I think one must be content with that even if it is impossible to see how it may be so. (§64)

Summary

In Anselm's preface to Monologion, he writes that he was asked by his fellow monks to make an argument not resting "on the basis of the authority of Scripture," but to use the "constraints of reason" to prove his claims. He admits the difficulty of the task, and his own limitations, but the fact that he wrote the book the way he did indicates that he thought it profitable to argue in this manner.

Furthermore, he believes he expresses orthodox doctrine: he says he has "been unable to find anything which is inconsistent with the writings of the Catholic Fathers, and in particular with those of the Blessed Augustine." This is a fair assessment: as mentioned, his analogy of Consciousness, Understanding, and Love originally comes from Augustine, and he is careful to emphasize both the unity and the plurality of the Trinity, thus avoiding the heresies of Sabellianism, tritheism, and Arianism.

Quotations here come from the 2008 edition of Anselm of Canterbury: The Major Works in the Oxford World's Classics series.

Public domain editions of Proslogion and Monologion are widely available, but are not based on modern critical editions of Anselm's works, and thus may not closely match the quotes provided here.