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If this feels like 4/4 time with an extra beat tagged on at the end, then I would count each measure as:

1-ee-and-uh-2-ee-and-uh-3-ee-and-uh-4-ee-and-uh-5`

If you want the pattern to count up and don't need to worry about accenting a particular beat, I would recommend this:

1-ee-and-uh-IS-ee-and-uh-SEV-ee-and-uh-EN-ee-and-uh-TEEN`
2-ee-and-uh-IS-ee-and-uh-SEV-ee-and-uh-EN-ee-and-uh-TEEN`
3-ee-and-uh-IS-ee-and-uh-SEV-ee-and-uh-EN-ee-and-uh-TEEN`

Then, internalize the "ee-and-uh"s so that you don't have to count them out loud. This leaves you with:

1 / / / IS / / / SEV / / / EN / / / TEEN`
2 / / / IS / / / SEV / / / EN / / / TEEN`
3 / / / IS / / / SEV / / / EN / / / TEEN`

If you were to learn this method, it would be easier because you'll only have to count 5 beats, and the rest you will be able to simply feel intuitively.

Another value of this pattern is that, if you are learning to count/feel lots of different odd meters, then you can easily remember which one is 17/8 versus, e.g., 15/8.

If you want the 17 beats in groupings of 3 (so that every third beat is a strong beat), then you could use:

1-and-uh-THIS-and-uh-IS-and-uh-SEV-and-uh-EN-and-uh-TEEN-OH
2-and-uh-THIS-and-uh-IS-and-uh-SEV-and-uh-EN-and-uh-TEEN-OH
3-and-uh-THIS-and-uh-IS-and-uh-SEV-and-uh-EN-and-uh-TEEN-OH

Once you internalize the weak beats (the "and-uh"s), you can count it more easily like this:

1 / / THIS / / IS / / SEV / / EN / / TEEN OH
2 / / THIS / / IS / / SEV / / EN / / TEEN OH
3 / / THIS / / IS / / SEV / / EN / / TEEN OH

Assuming captial 'Da' is the stress. It's duple and 16+1.

But, a meter like 17/8 seems to obscure the very regular stress given except for the last group.

It seems a mix of 2/4 and 5/8 meters makes the clearest break down of beats and stresses.

2/4 1 + 2 + | 1 + 2 + | 1 + 2 + | 5/8 1 + 2 + a

I'm seeing this as a subdivision of 2 rather than 4. So instead of treating 'Da-da-Da-da' as 1 e & a I reduce that to just Da-da twice which counts as 1 + 2 +. If you had written something like Da-da-ti-da or otherwise differentiated the 1st and 3rd events, then a subdivision of 4 would have been appropriate.

I don't see how the various mnemonics suggested in other answers are supposed to help.

If it's put into meters that reflect your stress pattern, it's not overly complicated, and a mnemonic becomes necessary. Personally, I feel the mnemonic seems to obscure the pattern.

If a mnemonic is really desired, wouldn't it make sense to pick parts of speech to reflect the beat divisions?...

up and down and
left and right and
in and out and
there and back a-gain

... that a pattern of 4 + 4 + 4 + 5 = 17 eighth notes.

Sheet music and notes would make the question more clear

Using 4 times Da-ba-da-ra with a final Da and counting with the fingers of the left hand instead of 1234 - 2234 - 3234 4234 5!) and the right hand counts the lines would make it easy to get along : 4 + 4 + 4 + 4 + 1=17

1) Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da

2) Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da

3) Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da-ba-da-ra + Da

In jazz counting 8th notes goes Da-ba-Da-ba
(where the -ba is creating a swing as the da is longer than the ba.)
Thus you might also use this by counting 4 times Da-ba-Da-ba with a final Da! for each line.

What is some-thing sim-il-ar that would fit for three groups of sev-en-teen.

Not sure if I understand where you want to put the accents. If it does not matter then I think that I would break down 17 into a nine plus an eight and then further break down the nine to three sets of three and the eight to four sets of two.

So my result would be

Da da da Da da da Da da da Da da Da da Da da Da da

or if counting

1 2 3 |2 2 3 |3 2 3 |1 2 |2 2 |3 2 |4 2

For me this is easier to count in my head or to say out loud.

Using konnakol (you can find many videos on it):

|| ta ka ta ki ṭa | ta ka ta ki ṭa | ta ka ta ki ṭa | taam . ||
|| ta ka ta ki ṭa | ta ka ta ki ṭa | ta ka ta ki ṭa | tohm . ||
|| ta ka ta ki ṭa | ta ka ta ki ṭa | ta ka ta ki ṭa | dheem . ||

Quick note: The 't' is dental (close enough to the Spanish/French denti-alveolar 't') and 'ṭ' is retroflex (but can be approximated by alveolar 't' of English).

The period (.) indicates a rest (of unit duration)

Here the difference in the three phrases is at the end, but it could easily be moved to the start.